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Manual  of  Experiments 
in  General  Physics 


BY 

William  F.  Schulz,  Ph.  D. 

Assistant  Professor  of  Physics , University  of  Illinois 

With  Additional  Experiments  by 
E.  H.  Williams,  Ph.  D. 

Associate  in  Physics 9tr.  University  of  Illinois 

SEP  1 9 1919 


Printed  for  Sophomore  students  in  the  laboratory  courses 
in  general  physics,  University  of  Illinois 

(Not  Published) 


Rogers  Printing  Company 
Dixon  and  Chicago,  Illinois 


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Preface-  3 /5ig 

This  manual  is  written  for  the  use  of  sophomore  students 
in  the  general  physics  courses  at  the  University  of  Illinois. 
The  directions  are  such  as  apply  to  the  particular  forms  of 
apparatus  and  facilities  to  be  found  in  this  laboratory. 
The  methods  of  performing  experiments,  and  the  designs  of 
special  forms  of  apparatus  have  been  developed  under  the 
supervision  of  Professor  A.  P.  Carman,  Director  of  the 
Physical  Laboratory.  In  the  second  edition  of  this  manual, 
Dr.  E.  H.  Williams,  Associate  in  Physics,  has  added  or  has 
re-written  experiments  Nos.  4,  6,  10,  18,  19,  20,  22,  25,  30, 
48  and  55.  The  laboratory  course  runs  parallel  with  the 
lecture  course  in  general  physics.  The  experiments  are  per- 
formed in  groups  of  three,  which  are  more  or  less  related, 
but  independent  so  far  as  the  order  in  which  they  are  per- 
formed is  concerned.  With  ten  sets  of  each  apparatus,  a 
class  of  sixty  students  is  provided  for  in  the  laboratory  at 
one  time,  by  letting  students  work  together  in  pairs  and 
having  the  three  related  experiments  in  operation  simul- 
taneously. The  experiments  to  be  performed  during  the 
first  semester  are  described  in  detail;  the  directions  for  those 
designed  for  the  second  semester’s  work  are  made  less 
definite,  leaving  the  student  some  choice  in  the  method  of 
procedure,  or  giving  him  a chance  to  exercise  his  ingenuity 
and  making  him  more  dependent  upon  his  own  knowledge 
of  the  underlying  principles. 

Material  for  the  tables  in  the  appendix  has  been  drawn 
from  a large  number  of  sources,  and  it  would  be  difficult 
to  give  credit  to  all  of  these,  hence  such  references  have  been 
omitted  entirely. 

Physics  Laboratory, 

University  of  Illinois 

June  30,  1916. 


CONTENTS 


Introduction 

page 

Laboratory  Directions  .....  9 

Units  ........  12 

Significant  Figures  ...  . . . .14 

Errors  . . . . . . , .15 

The  Plotting  of  Curves  .....  20 

Experiments  in  Mechanics 
no. 

1 —  Density  of  Solids  .....  27 

2 —  Testing  a Spirit  Level  .....  33 

3 —  Equilibrium  of  Concurrent  Forces.  Composition 

and  Resolution  of  Forces  ....  37 

4 —  Equilibrium  of  Concurrent  Forces  by  the  Triangle 

of  Forces  ......  40 

5 —  Equilibrium  of  Non-Concurrent  Forces  . . 44 

6 —  Parallel  Forces  ......  49 

7 —  The  Fine  Balance  .....  52 

8 —  The  Pendulum  . ....  58 

9 —  The  Law  of  Falling  Bodies  . . . 66 

10 —  Acceleration  of  Gravity  with  a Falling  Tuning  Fork  69 

11—  Friction 72 

12 —  Stretching  Wires  and  Young’s  Modulus  . . 77 

13 —  Laws  of  Bending  of  Rods.  Modulus  of  Elasticity  82 

14 —  Laws  of.  Twisting  of  Rods.  Modulus  of  Rigidity  87 

15 —  Moment  of  Inertia  . ....  90 

16 —  The  Torsion  Pendulum  . . . . . 95 


17 —  The  Spiral  Spring  Balance.  Specific  Gravity  . 100 

18 —  Specific  Gravity  with  Hydrometers  . . . 105 

19 —  Specific  Gravity  by  Hare’s  Method  . . 108 

20 —  Surface  Tension : 

Part  1. — By  Capillary  Tubes  . . . Ill 

Part  2. — By  Means  of  the  Jolly  Balance  . 114 

21 —  Boyle’s  Law  ....  . . . 116 

Experiments  in  Heat 

22 —  Calibration  of  Thermometers  . . . . 121 

23 —  Charles’  Law  . . . . . .126 

24 —  Linear  Expansion  . . . . . .131 

25 —  Coefficient  of  Cubical  Expansion  of  a Liquid  . 134 

Calorimetry  . . . . L . .138 

26 —  Specific  Heat  . . . . . 139 

27 —  Mechanical  Equivalent  of  Heat  . . .142 

28 —  Latent  Heat  of  Fusion  .....  147 

29 —  Latent  Heat  of  Vaporization  . . . .149 

30 —  Hygrometry,  Humidity  and  Dew  Point  . . 152 

Experiments  in  Magnetism  and  Electricity 

31 —  Magnetic  Fields  of  Force  . . . .157 

32 —  Action  Between  Magnetic  Poles  . . . 162 

33 —  Determination  of  H.  and  M.  . . . . 168 

34 —  Galvanometers  . . . . . . 173 

35 —  Ohm’s  Law  and  Potential  Drop  . . . 184 

36 —  The  Wheatstone  Slide  Wire  Bridge  . . . 190 

37 —  The  Post  Office  Box  Bridge  . . . .199 

38 —  The  Potentiometer  . . . . 204 

39 —  The  Figure  of  Merit  of  a Galvanometer  . . 213 

40 —  Internal  Resistance  of  a Galvanic  Cell  by  Ohm’s 

Method  .......  217 

41 —  Potential  Difference  at  the  Terminals  of  a Cell  as 

a Function  of  the  External  Resistance  . . 218 


42 —  Electrolysis  and  the  Copper  Voltameter  . 224 

43 —  Electromagnetic  Induction  ....  229 

44 —  Joule’s  Law  and  the  Electro-Calorimeter  . . 233 

Experiments  in  Light 

v 

45 —  Photometer  .......  237 

46 —  Reflection  and  Mirrors  .....  242 

47 —  Refraction  and  Lenses  . . - . . 252 

48 —  Index  of  Refraction  of  a Lens  . . : 258 

49 —  Critical  Angle  and  Index  of  Refraction  of  Glass 

in  Air  .......  265 

50 —  Microscopes  and  Telescopes  ....  269 

51 —  The  Spectrometer  ......  273 

52 —  The  Diffraction  Grating  .....  277 

Experiments  in  Sound 

53 —  Graphical  Determination  of  Pitch  of  Tuning  Forks  284 

54 —  Vibrations  of  Stretched  Strings  . . . 286 

55 —  Resonant  Air  Columns  .....  288 

Appendix 

I — Common  Conversion  Factors  . . . 292 

II — Specific  Gravity  .....  293 

III —  Elastic  Constants  . . . . 293 

IV —  Moments  of  Inertia  .....  294 

V — rBarometer  Corrections  . . . . 294 

VI — Heat  Constants  - . . . . 295 

VII — Steam  Temperatures  . . . f 296 

VIII — Hygrometry  Table  .....  296 

IX — Specific  Resistance  .....  297 

X — Electrochemical  Equivalents  . . . 297 

XI — Indices  of  Refraction  .....  297 

_ _ + 

XII — Natural  Trigonometric  Functions  . . 298 

XIII — Logarithms  .....  299-300 


Introduction. 


Laboratory  Directions. — In  order  to  facilitate  the  dis- 
tribution of  apparatus,  handling  of  large  classes,  etc.,  the 
following  method  of  procedure  is  in  use. 

The  experiments  are  performed  in  groups  of  three,  the 
classes  being  divided  into  three  groups,  and  the  particular 
experiment  to  be  performed  by  each  group  is  posted  on  the 
bulletin  board. 

Experiments  are  performed  in  cyclic  order;  that  is,  group 
one  will  perform  the  experiments  in  the  order  1-2-3,  4-5-6, 
etc.,  group  two  in  the  order  2-3-1,  5-6-4,  etc.,  group  three 
in  the  order  3-1-2,  6-4-5,  etc.,  the  order  remaining  the  same 
throughout  the  semester. 

Students  generally  work  in  pairs,  the  assignment  of 
partners  being  made  at  the  beginning  of  each  period.  Stu- 
dents are  required  to  read  carefully  the  description  of  the 
assigned  experiment  so  that  they  are  prepared  to  perform  it 
without  referring  to  the  manual. 

The  first  ten  or  fifteen  minutes  of  each  period  are  devoted 
to  a discussion  of  the  experiment  to  be  performed.  The 
instructor  calls  attention  to  those  parts  of  the  experiment 
which  require  particular  care,  and  students  are  given  a 
chance  to  ask  questions  concerning  those  points  which  they 
do  not  understand. 

At  the  beginning  of  a laboratory  period,  each  student 
and  the  partner  assigned  him  for  that  particular  period  will 
be  given  a key  to  an  apparatus  locker,  and  three  blank  data 
sheets  on  which  will  be  written  the  number  of  the  experiment 
assigned,  the  number  of  the  apparatus  locker  and  the  place 
of  working.  The  laboratory  manual  gives  a list  of  appa- 


10 


Manual  of  Experiments 


ratus  needed,  and  the  directions  for  performing  the  experi- 
ment. This  list  will  be  found  on  the  data  sheets  also.  The 
student  will  inspect  the  apparatus  in  the  locker  and  at  the 
place  of  working,  notifying  the  instructor  at  once  of  missing 
or  damaged  apparatus.  Students  are  held  responsible  for 
apparatus  assigned  them.  The  experiment  will  be  per- 
formed in  accordance  with  the  laboratory  manual,  and  the 
results  obtained  recorded  on  the  data  sheets.  The  measure- 
ments taken  may  be  recorded  first  in  pencil  on  one  of  the 
sheets,  but  must  be  copied  in  ink  on  the  others.  The  nu- 
merical calculations  should  be  completed  in  the  laboratory 
so  that  the  instructor  may  point  out  any  errors. 

Apparatus  must  be  returned  to  the  locker,  and  the  place 
of  working  be  left  clean.  The  students  will  then  return  the 
locker  key  and  data  sheets  to  the  instructor,  who  will  stamp- 
them  and  return  the  ink  copies  to  be  bound  in  the  written 
report  of  the  experiment. 

Alternate  periods  will  be  devoted  to  performing  an  ex- 
periment and  writing  a report  of  it,  until  the  student  has 
learned  how  to  write  the  reports  properly.  After  the  first 
few  weeks,  three  experiments  will  be  performed  in  successive 
laboratory  periods,  and  followed  by  a written  report  of  only 
one  of  them.  The  results  of  calculations  with  the  data 
taken,  and  answers  to  questions  and  problems,  are  however 
always  due  at  the  beginning  of  the  period  following  that  on 
which  the  experiment  was  performed.  In  any  case,  each 
set  of  experiments  will  be  followed  by  a written  quiz  on  all 
three  of  them. 

Writing  of  Reports: — The  report  of  each  experiment 
is  to  be  written  in  ink  and  bound  in  one  of  the  flexible  covers 
used  for  Engineering  College  reports. 

The  title  page  should  be  properly  filled  out  in  free  hand 
lettering. 


In  General  Physics 


11 


The  data  sheet  should  precede  the  report  and  when  a 
curve  is  plotted  it  should  be  placed  face  to  face  with  the 
data  sheet.  The  curve  must  be  drawn  according  to  the 
directions  given  at  the  end  of  this  introduction. 

In  writing  up  reports  the  following  topics  should  be 
treated  in  logical  order — (a)  Object  of  experiment;  (b) 
Theory;  (c)  Method;  (d)  Conclusions  and  discussion; 
(e)  Questions  and  problems. 

Under  “Object  of  experiment ” give  a brief  and  accurate 
statement  of  what  you  think  is  the  object  of  the  experiment. 

Under  “ Theory  ” give  briefly  the  fundamental  theory 
underlying  each  experiment  and  definitions  of  the  physical 
quantities  which  have  been  determined,  as  well  as  the  equa- 
tions used  in  calculating  results. 

Under  “Method”  draw  (in  ink  and  with  instruments) 
a diagram  of  the  essential  parts  of  the  apparatus  used 
whenever  this  will  aid  materially  in  describing  the  experi- 
ment, and  give  an  abstract  of  the  process  of  performing 
the  experiment.  Avoid  the  use  of  personal  pronouns. 

Under  “Conclusions  and  Discussion”  tell  briefly  but 
accurately  all  that  you  have  learned  about  the  phenomena 
illustrated  by  the  experiment,  and  discuss  the  accuracy  of 
your  results,  stating  the  sources  and  magnitude  of  errors 
which  may  occur  and  methods  of  avoiding  them,  as  well 
as  the  magnitude  of  the  errors  made  by  neglecting  any  fac- 
tors in  the  computation. 

Answer  the  questions  and  solve  the  problems  assigned  for 
each  experiment,  indicating  the  method  of  solving  and 
giving  the  final  result. 

References: — References  will  be  given  to  sections  in 
the  following  books: 

D. — Duff’s  Textbook  of  Physics,  Third  edition. 

G. — Ganot’s  Physics  (Atkinson)  Seventeenth  edition. 


12 


Manual  of  Experiments 


K. — Kimball's  College  Physics. 

W. — Watson's  Text  book  of  Physics,  Fourth  edition. 

Units: — The  magnitude  of  every  physical  quantity  ob- 
tained by  measurement  or  by  calculation  must  include  the 
unit  in  which  this  quantity  is  expressed,  and  a numeric 
giving  the  number  of  times  the  unit  is  contained  in  the 
quantity.  It  is  evident  that  we  can  have  no  idea  of  the 
magnitude  of  a physical  quantity  in  general  unless  it  is 
expressed  in  terms  of  a definite  unit.  Thus  to  say  that  a 
length  is  two  conveys  no  idea  of  its  size  unless  we  state 
whether  it  is  two  inches  or  two  miles,  etc. 

On  the  other  hand  some  physical  constants  are  ratios 
between  two  physical  quantities  of  the  same  kind.  For 
example,  the  specific  gravity  of  a substance  is  the  ratio  of 
the  density  of  the  substance  to  the  density  of  water,  and  is 
expressed  as  an  abstract  number  without  a specific  unit 
being  mentioned.  Specific  heat  of  a substance  is  the  ratio 
of  the  quantity  of  heat  required  to  raise  one  gram  of  the 
substance  one  degree  Centigrade  at  a definite  temperature 
to  the  quantity  required  to  raise  one  gram  of  water  one 
degree  at  the  same  temperature,  and  a number  is  all  that 
is  necessary  to  express  this  ratio. 

Systems  of  Units: — D.  149,  150,  154. — All  the  units  of 
Mechanics  may  be  expressed  in  terms  of  three  independent 
units  called  “fundamental  units."  All  other  units  are  de- 
fined by  reference  to  these  fundamental  units  and  are  called 
“derived  units." 

A system  of  units  in  which  the  derived  units  bear  the 
simplest  possible  relation  to  the  fundamental  units  is  called 
“an  absolute  system." 

The  three  fundamental  units  in  general  use  are  those 
of  length,  mass  and  time.  In  the  C.  G.  S.  absolute  system, 
the  centimeter,  gram  and  second  are  the  fundamental  units. 


In  General  Physics 


13 


In  the  F.  P.  S.  absolute  system,  the  foot,  pound  and  second 
are  the  corresponding  fundamental  units. 

A third  system  of  units,  sometimes  called  the  gravi- 
tational system,  is  frequently  used.  In  the  absolute  sys- 
tems, the  unit  of  force  is  defined  as  the  force  which  gives 
unit  acceleration  to  the  unit  mass.  The  C.  G.  S.  unit  of 
force  is  called  the  dyne,  the  F.  P.  S.  unit  the  poundal.  In 
the  gravitational  systems  the  unit  of  force  is  the  weight  of 
unit  mass,  or  the  force  with  which  the  earth  attracts  the 
unit  mass,  giving  it  the  acceleration  of  gravity,  11  g.  ’ ’ 
Since  the  gravitational  unit  of  force  gives  to  unit  mass  in 
unit  time  “gM  times  as  much  acceleration  as  does  the 
absolute  unit,  the  gravitational  unit  of  force  must  equal 
the  absolute  unit  multiplied  by  the  acceleration  of  gravity 
“gM.  The  gram  weight  is  981  dynes  where  g = 981 
cm. /sec2.  The  pound  weight  is  32  poundals  where  g = 32 
ft.  /sec2. 

In  expressing  the  result  of  a measurement  or  calculation 
the  following  directions  should  be  followed: 

Be  sure  to  state  the  unit  in  which  the  result  is  expressed. 

Unless  otherwise  directed  use  the  centimeter,  the  gram 
and  the  second  as  the  unit  of  length,  of  mass  and  of  time 
respectively. 

Express  all  fractions  in  the  decimal  form. 

To  avoid  repetition  of  cyphers,  express  all  large  numbers 
in  positive  powers  of  ten,  all  small  numbers  in  negative 
powers  of  ten.  For  example, 

30,000,000,000  = 3 x 1010.  0.0000117  = 11.7  x 10“6 

Be  careful  to  distinguish  between  unit  mass  and  unit 
weight  (used  as  a force)..  Use  the  symbols  gm.  and  lb.  for 
gram  and  pound  mass  respectively,  the  symbols  gm.  wt.  and 
lb.  wt.  for  gram  and  pound  weight  respectively. 

Where  a unit  has  not  been  given  a name,  express  it  in 


14 


Manual  of  Experiments 


terms  of  fundamental  units.  For  example,  in  the  C.  G.  S. 
system,  the  unit  of  velocity  is  one  cm.  per  sec.  (cm. /sec.); 
unit  acceleration,  one  cm.  per  sec.  per  sec.  (cm. /sec2.)  etc. 

Significant  Figures: — The  accuracy  of  physical  mea- 
surements depends  amongst  other  things  upon  the  instru- 
ments or  apparatus  used.  For  instance  in  measuring  a 
length,  the  vernier  is  correct  to  0.01  cm.  and  the  micrometer 
to  0.001  cm.  But  it  is  possible  to  estimate  fractional  parts 
of  a scale  division  on  these  instruments,  so  that  we  can  ex- 
press the  length  to  one  more  decimal  place  than  that  indi- 
cated above,  although  this  last  figure  is  somewhat  in  doubt. 
It  is  customary  to  retain  this  last  figure  in  the  measurement. 
If  however  all  the  doubtful  figures  are  retained  in  calcula- 
tions with  these  measurements,  we  shall  waste  considerable 
time  in  unprofitable  calculations  and  also  give  a false  idea 
of  the  accuracy  of  the  final  result. 

For  instance  if  we  are  determining  the  area  of  a rec- 
tangle and  the  measurement  of  the  sides  gives  us  4.35  and 
11.51  respectively,  the  last  figure  in  each  being  doubtful, 
(which  is  indicated  here  by  underscoring  them),  the  product 
is  50.0685,  but  the  underscored  0 and  all  the  figures  after 
it  result  from  multiplication  with  a doubtful  figure,  and  are 
therefore  doubtful  themselves.  These  figures  should  there- 
fore be  discarded  in  any  further  calculations,  to  save  time, 
the  area  being  written  50.1  sq.  cm.  If  retained  they  would 
give  the  false  impression  that  the  area  as  calculated  is 
correct  to  the  ten  thousandth  decimal  place. 

Again  if  we  are  determining  the  density  of  a body  (or 
mass  per  unit  volume)  and  find  its  mass  and  volume  to  be 
49.2gm.  and  4.352cc.  respectively,  the  quotient  of  these  two 
will  give  11.305,  but  the  underscored  3 and  the  following 
figures  are  the  result  of  operations  with  doubtful  figures  and 


In  General  Physics 


15 


are  themselves  doubtful,  so  that  the  density  should  be 
written  11.3  gm./cc. 

All  figures  up  to  and  including  the  first  doubtful  figure 
are  called  significant  figures. 

It  is  readily  seen  from  these  examples  that  the  number 
of  significant  figures  in  the  final  result  is  the  same  as  the 
least  number  of  significant  figures  in  any  one  of  the  factors. 

If  the  second  doubtful  figure  is  less  than  five  it  is  to  be 
discarded.  If  it  is  five  or  more  the  first  doubtful  figure 
is  to  be  increased  by  one,  as  in  the  first  example  given. 

In  computations  with  logarithms,  as  many  places  should 
be  retained  in  the  mantissae  as  there  are  significant  figures 
in  the  data.  Thus  log  3.1416  = 0.49714.  The  characteristic 
is  not  to  be  considered  in  counting  the  figures. 

When  the  purpose  of  a zero  is  simply  to  fix  the  decimal 
place  in  front  of  significant  figures  it  is  not  counted  as  a 
significant  figure.  Thus  .0345,  1.03,  and  32.0  each  have 
three  significant  figures. 

Where  the  omission  of  the  second  doubtful  figure  would 
make  a difference  as  great  as,  or  greater  than  the  experi- 
mental error,  it  should  be  retained. 

Errors: — Since  the  accuracy  to  which  physical  measure- 
ments are  made  depends  amongst  other  things  upon  the 
construction  of  the  instruments  used,  there  is  always  a 
calculable  error  in  every  measurement.  Thus  a micrometer 
may  indicate  a length  correctly  to  0.001  cm.,  but  by  estimat- 
ing the  fractional  part  of  a scale  division,  the  length  may  be 
expressed  to  0.0001  crm  If  the  scale  divisions  are  sufficiently 
large  and  the  eye  of  the  observer  sufficiently  trained,  the 
measurement  may  be  correct  to  0.0001  cm.  Usually  the 
readings  with  the  particular  micrometer  calipers  to  be  used 
are  correct  to  only  0.0003  cm.  or  even  0.0005  cm.  The 
possible  error  in  the  length  from  this  source  alone  would 


16 


Manual  of  Experiments 


then  be  0.0005  cm.  and  a length  X measured  with  such  a 
degree  of  accuracy  would  be  recorded  0.0005  cm.  It  is 
sometimes  desirable  to  know  the  effect  that  such  errors 
have  on  the  final  result  of  computations  with  these  measure- 
ments. We  shall  develop  approximate  rules  for  the  cal- 
culation of  such  errors. , 

(i a ) — Rule  for  sums  and  differences . — If  the  only  operations 
performed  with  the  measurements  are  addition  and  subtraction , 
the  maximum  possible  error  in  the  final  result  will  be  the  sum 
of  the  errors  in  the  measurements . For  instance  if  two 
measurements  X and  Fhave  errors  =*=  a and  =*=  b respectively, 
their  sum  will  be  (X-\-Y)  =±=  (a=*=b)  and  their  difference 
(X—  Y)  ± (a=Fb).  In  each  case  the  maximum  error  is  the 
sum  of  a and  b,  the  minimum  error  their  difference. 

If  now  the  operations  involve  multiplication  or  division, 
it  is  convenient  to  use  the  percent  error  rather  than  the 
numerical  error.  A percent  error  means  the  error  per  hun- 
dred parts.  Thus  an  error  of  2 parts  in  fifty  is  an  error  of 
4 parts  in  100  or  4%.  An  error  of  0.004  parts  in  0.020  is 
(0.004/0.020)  X 100  or  20%.  The  percent  difference  be- 
tween readings  of  the  same  quantity  is  calculated  from  the 
mean  or  average.  Thus  if  two  measurements  give  49  and 
51  units  respectively,  their  mean  is  (49  + 51)/2  = 50,  their 
difference  51—49  = 2,  the  part  difference  is  2/50  and  the 
percent  difference  (2/50X100)  =4%. 

The  error  from  the  mean  is  sometimes  asked  for.  This 
is  the  difference  between  any  one  reading  and  the  mean. 
In  the  case  of  two  numbers  the  percent  error  from  the  mean 
of  the  two  readings  is  the  same,  namely 


50-49 

50 


51-50 

x 100,  or  — 

' 50 


x 100  = 2% 


Or  in  general,  if  X and  Y be  the  numbers,  the  % error 


In  General  Physics 


17 


from  the  mean  will  be 

X-Y  X+Y  X-Y 

— — _ _i_  _ x 100,  or  — x 100. 

2 2 X+Y 

(b) — Rule  for  products  and  quotients.: — The  maximum 
possible  percent  error  in  a product  or  in  a quotient  is  the  sum 
of  the  percent  errors  in  the  separate  factors  provided  all  of  the 
errors  tend  to  increase  or  all  tend  to  decrease  the  product  or 
quotient.  That  is,  if  u,  v,  and  w are  respectively  the  percent 
errors  in  X,  Y,  and  Z,  then  the  error  in  any  multiplication  or 
division  between  these  factors  (as  for  instance  XY/Z)  will 
be  ( u+v+w ) %,  under  the  condition  mentioned.  (It  should 
be  stated  that  this  rule  is  only  approximate,  but  that  it  gives 
a good  idea  of  the  maximum  possible  per  cent  error  and  saves 
considerable  calculation  that  would  be  necessary  if  the 
numerical  errors  were  carried  through  the  whole  calculation.) 

To  derive  this  rule  let  the  measurements  X and  Y have 
the  numerical  errors  =±=  a and  =±=  & respectively  multiplying 
(X+a)  by  (Y+b)  we  get  {X+a)(Y +b)  = XY  +aY +bX+ab. 
a and  b are  small  quantities  and  their  product  ab  is  usually 
a very  small  fraction  which  may  be  neglected  in  comparison 
with  aY  and  bX. 

The  error  in  this  product  is  therefore  aX+bY.,  for  if 
X and  Y were  accurate,  their  product  would  be  XY.  The 
fractional  error  is  therefore  the  difference  between  the 
calculated  and  true  values  of  the  product  divided  by  the 
true  value,  or  (aY+bX)/XY  = a/X+b/Y.  The  percent 
error  is  this  fraction  of  100.  The  same  result  is  obtained 
when  the  negative  sign  is  used  with  a and  b.  But  100  a/X 
is  the  percent  error  in  X and  100  b/Y  is  the  percent  error  in 
Y.  Hence  the  percent  error  in  the  product  is  the  sum  of 
the  percent  errors  in  the  factors.  If  now  we  divide  (X+a) 
by  ( Y—b ),  (using  opposite  signs  so  that  both  errors  affect 


18 


Manual  of  Experiments 


the  quotient  in  the  same  way)  we  may  write  (X-\-a)/(Y—b)  = 
(X+a)  ( Y-b)-\ 

Expanding  the  last  term  and  neglecting  products  and 
powers  of  the  errors  which  become  very  small,  we  have 
(X+a)  (l/Y+b/Y2+etc:)=X/Y+a/Y+bX/Y*+etc. 

The  fractional  error  is  therefore  (a/Y+bX /Y2)  /{X /Y)  = 
a/X+b/Y  and  we  see  as  before  that  the  percent  error  in 
the  quotient  is  the  sum  of  the  percent  errors  in  the  numerator 
and  denominator.  The  same  result  is  obtained  when  the 
signs  of  a and  b are  interchanged. 

(c) — Rule  for  powers  and  roots — If  the  measurement  be 
raised  to  any  power,  the  percent  error  in  the  power  is  the 
percent  error  in  the  measurement  multiplied  by  the 
index  of  the  power,  i.  e.  if  squared,  the  percent  error 
is  multiplied  by  two,  for  the  square  of  a number  is  that 
number  multiplied  by  itself  and  by  rule  b)  the  percent 
error  in  a product  is  the  sum  of  the  percent  errors  in  the 
factors.  If  the  root  of  the  measurement  is  extracted,  the 
same  rule  applies,  for  the  square  root  is  the  measurement 
raised  to  the  Y/i  power,  etc.  Hence  the  rule  for  a number 
raised  to  a power  may  be  used,  and  if  u be  the  percent 
error  in  X.  the  percent  error  in  V X will  be  uf  2.  etc. 

Multiplying  or  dividing  the  measurement  by  a constant 
quantity  (in  which  there  is  no  error  of  measurement)  does 
not  alter  the  percent  error.  If  K be  a constant  and  X a 
measurement  with  a numerical  error  ±<2,  then  K (X=±a)  = 
KX^=Ka.  But  Ka  is  the  same  percent  of  KX  that  a is 
of  X.  The  same  reasoning  holds  if  we  multiply  by  1 /K,  or 
divide  by  K. 

If  on  the  other  hand  we  divide  the  constant  K by  the 
measurement  X with  an  error  of  b%,  since  the  error  in  K 
is  0,  the  error  in  the  quotient  is  by  our  rule  (o-\-b)%  or  the 
same  as  the  error  in  X. 


In  General  Physics 


19 


Percent  errors  are  also  useful  in  judging  the  relative 
importance,  so  far  as  the  final  result  is  concerned,  of  the 
errors  in  the  measurements  which  enter  as  factors  in  the 
calculation  of  this  result.  It  is  evident  that  a given  numeri- 
cal error  is  a larger  percent  of  a small  measurement  than  it 
is  of  a large  measurement.  For  instance  if  a micrometer 
caliper  is  accurate  to  0.0005  cm.  then  a measurement  of 
a length  of  5 mm.  made  with  it  will  have  a possible  error  of 
(0.0005/0.5)  X 100  or  0.1%  whereas  a measurement  of  10 
cm.  would  have  a possible  error  of  (0.0005/10)  X 100  = 
0.005%.  It  is  therefore  advisable  to  make  the  small  meas- 
urements more  carefully  than  larger  ones  and  with  instru- 
ments capable  of  greater  precision.  But  this  is  not  the  only 
thing  to  be  considered,  for  the  factors  in  a computation  do 
not  all  have  the  same  importance.  Some  of  the  factors  enter 
to  a higher  degree  than  others  and  it  should  be  remembered 
that  a measurement  which  is  cubed  has  its  percent  error 
multiplied  by  three,  etc.  Take  for  example,  the  volume 
of  a cylinder  of  length  / and  radius  r.  The  volume  is  wr2l , 
and  it  is  evident  that  the  relative  importance  of  the  errors 
in  r and  / depends  upon  their  magnitude  and  the  degree 
to  which  they  are  involved.  If  r and  / are  equal  and  are 
measured  with  the  same  instrument,  the  percent  error  will 
be  the  same  in  both,  but  since  r is  squared  while  l is  not,  the 
error  in  r has  twice  as  much  effect  upon  the  accuracy  of  the 
result,  as  the  error  in  /.  If  l = 2r  the  percent  error  in  r will 
be  twice  the  percent  error  in  / and  the  error  introduced  into 
the  calculated  volume  by  r2  will  be  four  times  that  due  to  /. 
If  r = 2l  the  percent  error  in  r will  be  3^2  the  percent  error 
in  / and  the  percent  error  in  r2  will  equal  that  in  /,  so  that 
both  are  of  equal  importance  in  the  calculation,  etc. 

In  this  way  the  relative  importance  of  the  different 
factors  in  a computation  may  be  studied.  It  is  always  ad- 


20 


Manual  of  Experiments 


visable  on  beginning  an  experiment  to  obtain  some  idea 
of  the  relative  magnitudes  of  the  measurements  made,  (by 
a preliminary  trial  if  necessary),  and  the  degree  to  which 
they  are  involved  in  the  computation,  to  determine  which 
should  be  measured  with  the  greatest  care  and  the  most  ac- 
curate instruments.  It  must  be  remembered  that  the  num- 
ber of  significant  figures  m the  result  of  the  computation 
depends  upon  the  least  number  of  significant  figures  in  any 
one  of  the  factors. 

To  avoid  useless  computation  it  is  advisable  to  inspect 
the  formulas  used  in  calculating  results  and  to  disregard  all 
terms  wdiose  omission  would  introduce  negligible  errors  only. 
For  instance  when  a shunted  galvanometer  is  used  in  series 
with  a large  resistance,  it  is  well  to  calculate  what  percent- 
age of  the  total  resistance  is  included  in  the  galvanometer 
and  shunt.  If  their  omission  introduces  a small  fraction  only 
of  one  per  cent  error  in  the  final  result,  the  term  for  the 
shunted  galvanometer  should  be  discarded  in  the  calculation. 

The  Plotting  of  Curves: — The  relation  between  two 
physical  quantities  which  vary,  the  one  with  the  other,  is 
often  more  clearly  seen  from  the  curve  or  graph  obtained  by 
plotting  on  coordinate  paper  a series  of  simultaneous  ob- 
servations of  the  two  quantities.  Such  a curve  has  other 
uses  besides  presenting  clearly  to  the  eye  the  relative  varia- 
tions of  the  two  quantities.  For  instance  we  may,  by  inter- 
polation, determine  corresponding  values  of  the  two  quan- 
tities, other  than  those  for  which  the  data  have  been  taken. 
Again  when  one  or  two  plotted  points  deviate  very  markedly 
from  a smooth  curve  drawn  through  the  remaining  points, 
we  may  discard  the  data  for  the  former  as  being  in  error. 

Usually  such  curves  are  plotted  on  rectangular  coordinate 
paper  having  centimeter  of  half  inch  spaces  divided  into 
tenths.  Sometimes  however  it  is  advantageous  to  plot  polar 


In  General  Physics 


21 


graphs  on  circular  and  radial  coordinates,  or  logarithmic 
curves  on  specially  ruled  logarithmic  paper.  In  the  follow- 
ing discussion  only  rectangular  coordinates  will  be  used. 
Two  intersecting  lines,  usually  the  left  hand  and  lower  edge 
of  the  ruled  space,  are  chosen  as  coordinate  axes,  the  hori- 
zontal one  being  called  the  axis  of  abscissae,  or  X,  the  vertical 
one  the  axis  of  ordinates,  or  Y.  To  plot  the  curve,  a measured 
value  of  the  independent  variable  is  laid  off  along  the  axis 
of  abscissae  from  the  intersection  of  the  axes.  From  this 
point  on  the  axis  the  corresponding  value  of  the  dependent 
variable  is  laid  off  parallel  to  the  axis  of  ordinates.  The 
point  reached  in  this  way  is  one  point  on  the  curve.  This  is 
repeated  for  each  pair  of  coordinate  values,  and  a smooth 
curve  is  drawn  through  the  plotted  points. 

As  a result  of  many  experiments  we  find  that  one  of  the 
measured  quantities  varies  directly  as  some  other,  or  the 
ratio  of  their  corresponding  values  is  a constant.  If  we  plot 
corresponding  values  of  such  quantities,  they  will  lie  along  a 
straight  line,  generally  inclined  at  an  angle  to  the  axes. 

The  general  equation  of  such  a curve  is 

y = a-\-bx 

where  y is  a value  of  the  ordinates,  x the  corresponding  value 
of  the  abscissae,  a is  a constant  representing  the  portion  of 
the  axis  of  ordinates  intercepted  by  the  curve,  and  b a con- 
stant denoting  the  slope  of  the  line.  As  an  example  we  may 
take  corresponding  values  of  the  mass  and  volume  of  a sub- 
stance. The  ratio  between  corresponding  values  of  mass 
and  volume  is  a constant,  the  density  of  the  substance,  hence 
in  symbols,  if  m represents  mass,  v volume  and  d density, 

m = dv. 

If  we  plot  corresponding  values  of  m and  v they  will  lie  along 
a straight  line,  inclined  at  an  angle  with  the  axis  of  x whose 


22 


Manual  of  Experiments 


tangent  is  d;  for  here  y = m , x = v , b = d;  also,  since  <z  = o,  the 
line  passes  through  the  zero  of  ordinates. 

As  a result  of  other  experiments,  it  is  found  that  one 
quantity  is  inversely  proportional  to  another,  so  that  the 
product  of  corresponding  values  is  a constant.  If  we  plot 
corresponding  values  of  such  quantities  they  will  lie  along 
a rectangular  hyperbola,  having  the  axes  as  asymptotes. 
The  equation  of  such  a curve  is 

yx  = c 

where  c is  a constant.  As  an  example  we  may  take  Boyle’s 
law  which  states  that  at  a constant  temperature,  the  volume 
of  a gas  varies  inversely  as  the  pressure,  or  in  symbols 

pv  — c 

Taking  y — p and  x = v,  the  following  corresponding  values 
of  p and  v are  plotted  in  figure  1,  (curve  A). 


p cm 

V CC. 

1 

V 

70 

50  . 0 

. 0200 

80 

43  . 7 

. 0229 

90 

38  . 9 

. 0256 

100 

35  . 0 

. 0285 

110 

31  . 8 

. 0314 

120 

29  . 1 

. 0343 

It  is  evident  that  one  can  see  more  readily  whether  a 
series  of  plotted  points  lie  on  a straight  line,  than  whether 
they  lie  along  an  hyperbola  or  other  curve,  and  also  that  the 
former  is  more  easily  plotted.  It  is  therefore  in  general 
advisable  to  draw  the  straight  line  whenever  possible.  In 
many  cases  one  can  tell  from  the  trend  of  the  plotted  points 


In  General  Physics 


23 


what  relation  exists  between  the  coordinates,  and  very  often 
it  is  possible  to  find  some  functions  of  the  variables  between 
which  a direct  ratio  exists.  On  plotting  corresponding  values 
of  these  functions  rather  than  of  the  variables  themselves, 
they  will  lie  along  a straight  line.  Thus  the  relation  between 
quantities  which  are  inversely  proportional  may  evidently 
be  expressed  as  a constant  ratio  between  one  of  these  and 
the  reciprocal  of  the  other,  so  that 

y = c{l/x) 


This  is  a linear  relation  between  y and  1/x,  and  on  plot- 
ting corresponding  values  of  y and  1/x  we  obtain  a straight 


24 


Manual  of  Experiments 


line.  In  figure  1 (curve  B)  this  is  shown  by  plotting  cor- 
responding values  of  p and  1/v. 

As  a further  example  we  may  take  the  expression  for  the 
strength  of  an  electric  current  in  terms  of  the  deflection 
which  it  produces  in  a tangent  galvanometer,  namely 

i = k tan  <j> 

where  i is  the  current  strength,  </>  the  deflection  and  k the 
reduction  factor  of  the  instrument,  a constant.  If  we  plot 
i and  </>  we  get  a curve.  We  may  judge  by  inspection  that 
it  is  a tangent  curve,  and  plot  corresponding  values  of  i and 
tan  0,  which  will  lie  on  a straight  line.  The  above  equation 
is  then  of  the  form  y = kx. 

As  another  example  of  this  process  we  may  take  the 
results  of  the  experiment  on  Joule’s  law.  Here  the  heating 
effect  of  an  electric  current  is  proportional  to  the  square 
of  the  current  strength,  or  in  symbols 

H = (i2R)/J 

where  H is  the  heat  energy,  i the  current  strength,  J the 
mechanical  equivalent  of  heat,  a constant,  and  R the  re- 
sistance of  the  circuit,  another  constant.  On  plotting 
H and  i,  we  get  a parabola,  but  corresponding  values  of  H 
and  i2  when  plotted  yield  a straight  line,  the  equation  being 
of  the  form  y^={R/J)x. 

Where  higher  powers  of  y and  x are  involved  in  the 
equation,  it  is  generally  advisable  to  proceed  in  a different 
manner  to  establish  a linear  relation  in  its  place,  namely,  'by 
taking  the  logarithms  of  both  sides.  Taking  the  general 
relation 

* . m n 

y = a x . 

where  m and  n are  any  powers  whatever,  and  passing  to 
logarithms,  we  have 

m log  y = log  a + n log  x 

which  is  a linear  relation  between  log  y and  log  x.  A simple 


In  General  Physics 


25 


example  of  this  is  the  one  already  given,  namely  Boyle’s 
law.  This  may  evidently  be  written 

p — c v~l 

or  in  logarithms 

log  p = log  c — log  V. 

If  now  we  plot  corresponding  values  of  log  p and  log 
c — log  v,  as  ordinates  and  abscissae  respectively,  we  shall 
find  that  the  plotted  points  lie  along  a straight  line. 

The  following  rules  should  be  observed  in  plotting 
curves : — 

Use  coordinate  paper  with  millimeter  or  1/20  inch 
spacings  on  a sheet  of  the  same  size  as  that  used  in  the  body 
of  the  report. 

In  order  to  have  the  curve  facing  the  data  sheet,  so  that 
we  may  refer  readily  from  one  to  the  other,  the  coordinate 
paper  should  be  laid  with  the  binding  edge  on  the  right  hand 
side  while  plotting  the  curve. 

Give  the  curve  a title,  which  where  possible,  indicates 
the  relation  between  the  two  plotted  quantities. 

Draw  two  heavy  black  lines  to  represent  the  coordinate 
axes,  and  on  each  of  these  mark  the  scale  and  name  of  the 
plotted  quantities . 

Plot  values  of  the  independent  variable  as  abscissae  and 
those  of  the  dependent  variable  as  ordinates. 

Make  the  value  of  a scale  division  correspond,  as  nearly 
as  possible,  to  the  least  quantity  which  you  can  measure. 
If  this  be  impossible  choose  such  values  for  scale  divisions, 
as  will  make  the  curve  most  nearly  fill  the  page  with  the 
observations  to  be  plotted. 

All  lettering  and  numbers  should  be  placed  in  such  a 
position  that  they  can  be  read  without  turning  the  paper. 

Around  each  plotted  point  as  a center  draw  a small 
circle  about  one  mm.  in  radius. 


26 


Manual  of  Experiments 


Through  and  between  the  plotted  points  draw  a smooth 
regular  curve,  such  that  if  the  points  do  not  all  lie  on  the 
curve,  there  will  be  approximately  as  many  on  one  side  as 
on  the  other,  alternately. 

Any  point  which  deviates  markedly  from  this  curve  may 
be  assumed  to  be  due  to  an  accidental  error  and  should  be 
discarded. 


In  General  Physics 


27 


Experiments  in  Mechanics 

1.  DENSITY  OF  SOLIDS 

The  object  of  this  experiment  is  to  determine  the  density 
of  various  solids  and  to  become  familiar  with  the  methods  of 
using  the  coarse  balance,  the  vernier  calipers  and  the  mi- 
crometer calipers. 

(References:  D.  162.  G.  10,  11,  27,  73.  K.  69,  159. 

W.  16-18,  95,  129.) 

Apparatus. — Coarse  balance  and  set  of  weights,  ver- 
nier calipers,  micrometer  calipers,  cylinders  of  wood,  brass 
and  steel. 

Description  and  Theory. — The  density  of  a substance 
is  defined  as  its  mass  per  unit  volume.  The  mass  of  the 
substance  is  to  be  found  by  comparison  with  known  masses 
on  a coarse  balance,  and  its  volume  to  be  calculated  from  its 
dimensions  measured  with  the  vernier  and  the  micrometer 
calipers. 


28 


Manual  of  Experiments 


an  accuracy  greater  than  0.1  gram  is  not  necessary.  It  is 
essentially  a lever  of  the  first  class  with  equal  arms,  and  when 
both  arms  are  not  equally  loaded  it  is  in  equilibrium  under 
the  action  of  two  equal  and  opposite  moments  about  the  ful- 
crum. It  consists  of  a beam  supported  on  knife  edges  at 
its  center,  with  a pan  supported  on  knife  edges  at 
each  of  its  ends.  (See  Fig.  2.)  Two  nuts  (N)  working  on 
a horizontal  screw  at  the  center  of  the  beam,  serve  to  bring 
it  into  its  equilibrium  position  when  there  is  no  load  in  either 
pan.  A vertical  pointer  (P)  at  the  center,  indicates  when 
this  balance  position  is  reached.  The  unknown  mass  in  one 
pan  is  balanced  as  nearly  as  possible  by  known  ‘ ‘ weights  ’ ’ 
placed  in  the  other  pan.  By  means  of  the  slider  (S)  moving 
along  a scale  on  the  front  of  the  beam,  the  known  weights 
can  be  adjusted  to  0.1  gram. 

The  vernier  calipers  consist  of  a straight  scale  having 
a fixed  jaw  at  right  angles  to  it  at  one  end,  and  a movable 
jaw  bearing  a vernier  scale.  The  vernier  is  a device  for 
measuring  fractions  of  a scale  division.  The  ordinary  ver- 
nier scales  have  a space  covering  N—  1 divisions  of  the  main 
scale,  divided  into  N parts,  so  that  each  vernier  division  is 


shorter  than  a main  scale  division  by  one  Nth.  of  the  latter. 
1/A  is  called  the  vernier  fraction.  When  the  two  jaws  of 


In  General  Physics 


29 


the  vernier  are  in  contact  the  zeros  of  the  two  scales  coincide. 
If  the  jaws  are  separated  until  the  first  mark  after  the  zero 
mark  on  one  scale,  coincides  with  the  first  mark  on  the  other, 
the  distance  between  the  two  zeros  (and  therefore  between 
the  two  jaws  also)  is  1/N  of  a scale  division.  If  separated 
until  the  second  marks  of  the  two  scales  coincide,  the  distance 
between  the  jaws  is  2/ N of  a scale  division;  if  the  third  marks 
coincide  3/N  of  a division,  etc.,  so  that  when  the  Nth.  marks 
coincide  the  jaws  are  separated  by  a whole  division.  To  de- 
termine any  distance  between  the  jaws,  for  instance  the 
length  of  an  object  placed  between  them,  we  note  the  whole 
number  of  divisions  on  the  main  scale  up  to  the  zero  of  the 
vernier.  If  this  zero  does  not  coincide  with  a main  scale 
division,  the  distance  is  greater  than  the  whole  number 
by  a fraction  of  a division  which  can  be  determined  by  ob- 
serving the  number  of  the  vernier  division  which  coincides 
(or  most  nearly  coincides)  with  a main  scale  division  (no 
matter  which  one.) 

In  figure  3,  the  vernier  reads  18.8  mm. 

The  micrometer  calipers  consist  of  a curved  or  rectangular 


frame,  one  end  of  which  serves  as  a fixed  jaw,  and  an  accurate 
screw  which  serves  as  a movable  jaw.  The  other  end  of  the 
frame  is^threaded  so  that  the  screw  can  be  turned  in  it, 


30 


Manual  of  Experiments 


toward  or  from  the  fixed  jaw.  It  bears  a straight  scale 
whose  divisions  correspond  to  the  pitch  of  the  screw.  The 
screw  is  turned  by  means  of  a milled  head  attach.ed  to  a 
sleeve  which  moves  over  the  straight  scale  and  which  has 
on  its  edge  a circular  scale  whose  equal  divisions  represent 
fractions  of  a turn.  Usually  the  pitch  of  the  screw  is  a half 
millimeter  so  that  one  turn  of  the  screw  advances  it  a half 
millimeter,  the  edge  of  the  sleeve  moving  over  one  division 
of  the  main  scale.  In  this  case  there  are  50  divisions  on  the 
circular  scale,  each  division  representing  one  fiftieth  of  a 
half  millimeter  of  0.01  mm.  In  some  cases  the  main  scale 
is  divided  into  millimeters,  instead  of  half  millimeters, 
each  division  corresponding  to  two  complete  turns  of  the 
screw.  The  object  to  be  measured  is  placed  between  the 
fixed  jaw  and  the  tip  of  the  screw.  When  these  two  are 
in  contact  the  zeros  of  the  two  scales  should  be  together. 
As  the  screw  is  turned  the  distance  between  them  is  in- 
dicated on  the  two  scales,  the  number  of  millimeters  on 
the  straight  scale,  the  fractional  part  of  a millimeter  on  the 
circular  scale. 

In  figure  4 the  reading  of  the  micrometer  scales  is  7.55 
mm. 

Directions. — Place  the  slider  of  the  balance  at  0 on  the 
scale  and  adjust  the  nuts  on  the  horizontal  screw  so  that  the 
balance  is  in  equilibrium,  i.  e.  the  pointer  swings  equidistant 
on  each  side  of  the  center.  Place  the  object  to  be  weighed  in 
one  pan  and  add  known  weights  to  the  other  until  the  beam 
is  very  nearly  balanced.  Then  the  slider  can  be  moved  over 
the  scale  until  equilibrium  is  again  established.  While  ad- 
justing the  weights  in  either  pan,  the- plunger  under  one  of 
them  should  be  held  between  thumb  and  fore  finger  in  such 
a way  as  to  form  a cushion  between  pan  and  support,  to  pre- 
vent any  jarring  between  them,  otherwise  the  knife  edges 


In  General  Physics 


31 


may  be  thrown  out  of  position  and  a readjustment  for  equi- 
librium will  be  necessary. 

The  slider  reading  is  to  be  added  to  the  known  ‘ 1 weights  ’ ’ 
when  they  are  in  the  right  hand  pan,  but  subtracted  when 
they  are  in  the  left  pan. 

Weigh  each  of  the  three  cylinders  to  0.1  gram  and  record 
their  masses  in  the  proper  place  on  the  data  sheet. 

Measure  the  length  of  each  cylinder  with  the  vernier  cal- 
ipers, and  its  diameter  with  the  micrometer  calipers  (if 
possible).  The  cylinders  may  not  be  perfectly  circular 
and  their  ends  may  not  be  perpendicular  to  their  axes  so 
that  measurements  of  the  same  dimensions  taken  at  different 
parts  of  the  same  cylinder  may  differ  slightly  from  each 
other.  To  get  a good  average  measurement,  the  values 
recorded  on  the  data  sheet  should  be  the  mean  of  about  10 
measurements  taken  at  different  parts  of  the  cylinder. 

To  measure  the  length  of  the  cylinder,  first  note  the  main 
scale  divisions  and  determine  the  vernier  fraction.  If  the 
zeros  of  both  scales  do  not  coincide  when  the  jaws  are  to- 
gether, note  the  zero  error  (the  distance  between  them) 
and  determine  whether  it  is  to  be  added  to  or  subtracted 
from  the  readings  to  be  taken.  The  cylinders  should  be 
placed  with  one- end  flat  against  the  inner  face  of  the  fixed 
jaw,  and  its  side  in  contact  with  the  main  scale,  and  the 
movable  jaw  should  then  be  moved  up  flat  against  the  other 
end,  just  touching  it,  and  clamped  in  position  while  the 
reading  on  the  scales  is  taken. 

To  take  a measurement  with  the  micrometer  calipers, 
the  scale  divisions  are  first  noted.  Then  the  jaws  are 
brought  together,  the  milled  head  being  held  between  the 
forefinger  with  slight  pressure  and  turned  until  it  just  slips 
between  the  fingers  instead  of  rotating.  After  noting  the 
zero  error  as  was  done  with  the  vernier  calipers,  the  cylinder 


32 


Manual  of  Experiments 


is  placed  between  the  jaws  and  in  contact  with  them,  the 
screw  being  turned  as  before  with  slight  pressure  until  it 
just  slips  through  the  fingers. 

Note: — Never  hold  the  milled  head  tight  while  turning 
the  screw  into  contact  as  this  will  jam  the  threads  and  injure 
them,  besides  causing  errors  in  the  determination. 

Care  should  be  taken  to  measure  the  diameter  and  not  a 
shorter  chord  of  the  cross  section  of  the  cylinder.  Where 
the  main  scale  is  divided  into  millimeters,  care  must  be  ob- 
served to  read  fractional  part  of  a division  correctly,  for  evi- 
dently a reading  on  the  circular  scale  may  be  so  many  hun- 
dredths of  a millimeter  or,  1/2  millimeter  plus  so  many  hun- 
dredths. In  this  case  observe  whether  the  edge  of  the  cir- 
cular scale  is  more  or  less  than  half  a division  from  the  last 
preceding  millimeter  mark.  If  this  can  not  be  determined 
by  eye,  remove  the  cylinder  from  between  the  jaws  and  turn 
the  screw  back  until  the  circular  scale  reads  zero.  Then 
observe  whether  the  edge  stands  at  the  millimeter  mark  or 
half  a millimeter  from  it. 

From  these  measurements  the  volume  and  the  density 
of  each  cylinder  are  to  be  calculated  and  recorded  on  the 
data  sheet. 

The  error  in  measurements  due  to  the  graduation  of 
scales  on  instruments  should  be  estimated  in  each  case  and 
recorded  on  the  data  sheet,  and  the  maximum  possible  per- 
cent error  in  volume  and  in  density,  due  to  these  errors  in 
measurements,  should  be  calculated. 

Questions: — 1 — Determine  from  an  examination  of 
your  calculations,  which  measurement  causes  the  largest 
error  in  the  density  of  each  cylinder,  and  explain  why. 

2 — A'  circular  scale  has  divisions  equal  to  1/6  of  a degree. 


In  General  Physics 


33 


A vernier  attached  to  it  has  a total  length  of  59  main  scale 
divisions,  and  is  itself  divided  into  30  parts.  Show  what 
the  vernier  fraction  is  in  this  case. 

2.  TESTING  A SPIRIT  LEVEL. 

In  this  experiment  we  are  to  determine  the  radius  of 
curvature  of  a level  tube  and  to  find  the  angle  of  tilt  necessary 
to  move  the  bubble  through  one  division  of  the  scale  on  the 
tube.  (D.  183.  G.  109.  K.  175.  W.  144.) 

Apparatus. — Level  tube,  level  tester  and  meter  stick. 

Description  and  Theory. — A spirit  level  is  an  instru- 
ment for  detecting  the  deviation  of  a plane  from  the  hori- 
zontal, and  for  measuring  small  angles.  It  is  a glass  tube 
having  its  inner  surface  accurately  ground  in  an  arc  of  a 
circle  with  a long  radius  of  curvature.  It  is  filled  with  a 
mixture  of  ether  and  alcohol  except  for  a small  bubble  of 
air  which  always  tends  to  move  toward  the  highest  point 
of  the  tube.  A scale  on  the  upper  surface  of  the  tube  has 
divisions  equal  to  1/10  inch.  When  the  tube  is  in  a hori- 
zontal position  the  bubble  is  in  the  center  of  the  tube, 
but  when  inclined  the  bubble  moves  through  a distance 
which  depends  upon  the  angle  of  inclination  and  the  radius 
of  curvature.  * 


/ 


D 

- L I J 

!-  -i::i 

TW 

S G 

Mil  lannnui  t 

in  ijnr 

ii Lj 

Fig.  5 


The  level  tester  consists  of  an  iron  arm  DG  hinged  at  one 
end  into  a heavy  base  and  carrying  a micrometer  screw  M 
at  the  other  end.  The  glass  level  tube  T is  mounted  in  V 
grooves  on  the  iron  arm  and  may  be  tilted  by  turning  the 


34 


Manual  of  Experiments 


micrometer  screw.  (See  Fig.  5.)  Attached  to  the  level 
tester  and  parallel  to  the  axis  of  the  screw  is  a millimeter 
scale.  A circular  disc  having  its  upper  edge  divided  into 
50  equal  divisions  is  attached  to  the  screw  and  moves  along 
the  straight  scale  as  the  screw  is  turned.  It  serves  to 
indicate  fractions  of  a turn.  The  pitch  of  the  screw  is  a 
half  millimeter,  so  that  each  of  these  divisions  represents 
0.01  mm. 


6 


In  figure  6,  let  BH  represent  the  inner  curved  surface  of 
the  tube  and  AC  the  vertical  radius  of  curvature  through 
the  highest  point  (center  of  the  bubble)  when  it  is  in  the 
horizontal  position,  A being  the  center  of  curvature.  If  the 


In  General  Physics 


35 


tube  is  inclined  at  an  angle  </>  with  the  horizontal  FD,  the 
bubble  moves  from  C to  B , and  the  vertical  radius  through 
its  center  will  be  AB.  The  radius  AC  makes  an  angle  </> 
with  its  former  position.  If  the  angle  <f>  be  measured  in 
radians  we  have 

cf)  = BC  — 5“  AB  (1) 

We  have  also  the  relation  0 = FG  -r-  GD,  or  if  </>  is  so  small 
that  the  arc  FG  and  the  distance  EG,  through  which  the  mi- 
crometer screw  raises  the  end  of  the  level  tester,  may  be 
considered  practically  equal, 

0 = EG  -5-  GD  (2) 

It  follows  from  (1)  and  (2)  that 

AB  = BC/4>  = BCXGD/EG  (3) 

» 

or  the  radius  of  curvature  equals  the  distance  the  bubble 
moves  multiplied  by  the  ratio  of  the  length  of  the  iron  arm 
to  the  vertical  height  through  which  its  end  is  raised.  Since 
the  bubble  moves  through  BC  divisions  when  the  tube  is 
tilted  through  an  angle  <£  radians,  the  angle  necessary  to 
move  it  one  division  will  be  6 = </>  -r-  BC,  but  by  equation  (1) 

0 = d>/BC  = 1 /AB  (4) 

when  AB  is  in  scale  divisions.  Therefore  the  reciprocal 
of  the  radius  of  curvature  gives  the  angle  of  tilt  necessary 
to  move  the  bubble  through  one  scale  division. 

It  should  be  noted  that  equation  (4)  can  be  true  only 
when  AB  is  expressed  in  scale  divisions,  hence  the  radius  of 
curvature  should  be  expressed  in  tenths  of  an  inch  for  this 
calculation. 

Directions. — Measure  carefully  the  length  of  the  iron 
arm  from  the  center  of  the  micrometer  screw  to  the  center 
of  the  hinge  with  a meter  stick.  As*  the  ends  of  a straight 
scale  become  worn,  the  end  divisions  are  usually  inaccurate 
and  measurements  should  be  taken  from  some  inside  division 
near  the  end.  The  length  should  be  estimated  to  1/10  mm. 


36 


Manual  of  Experiments 


Next  place  the  level  tube  in  the  V grooves  and  adjust  the 
micrometer  screw  until  one  end  of  the  bubble  is  just  under  a 
division  mark  at  the  end  of  the  scale.  Read  the  position  of 
both  ends  of  the  bubble  and  of  the  micrometer  and  record 
them  on  the  data  sheet.  Then  turn  the  micrometer  until 
the  end  of  the  bubble  is  at  some  division  near  the  opposite 
end  of  the  scale,  and  take  readings  as  before.  Repeat  this 
operation  five  or  six  times,  moving  the  bubble  from  one  end 
of  the  scale  to  the  other,  and  taking  readings  in  both  po- 
sitions, but  be  careful  to  move  the  end  of  the  bubble  over 
the  same  number  of  divisions  each  time  (or  between  the 
same  two  divisions).  Calculate  the  mean  displacement  BC 
of  the  center  of  the  bubble,  and  the  corresponding  mean 
height  EG. 

By  means  of  equation  (3)  calculate  the  radius  of  curva- 
ture and  express  it  in  meters  and  in  feet.  By  equation  (4) 
calculate  the  angle  of  tilt  for  a movement  of  one  scale  divi- 
sion and  express  it  in  radians  and  in  degrees.  (One  radian 
= 57°  17'  45"  = 206265") 

Note: — Be  careful  not  to  lean  on  the  table,  bench,  or 
apparatus,  nor  to  breathe  on  the  level  tube,  nor  to  handle  it 
more  than  is  absolutely  necessary  and  thus  avoid  errors  due 
to  a change  in  the  size  or  position  of  the  bubble.*  In  order 
to  diminish  the  error  due  to  lost  motion,  the  micrometer 
should  be  turned  so  as  to  raise  the  iron  strip  in  each  final  ad- 
justment before  taking  a reading. 

Questions.  1 — Which  is  the  more  sensitive,  a level 
with  a short  radius  of  curvature  or  one  with  a long  radius, 
and  why? 

2. — A carpenter’s  level  with  radius  of  curvature  of  30 
feet  is  placed  on  a board  10  feet  long  and  the  bubble  position 


In  General  Physics 


37 


noted.  The  spirit  level  is  then  reversed  and  the  bubble 
moves  1.5  cm  from  its  former  position.  How  much  in  cm. 
must  the  end  of  the  board  be  raised  to  level  it? 

3.  EQUILIBRIUM  OF  COPLANAR  CONCURRENT 
FORCES.  COMPOSITION  AND  RESOLUTION  OF 

FORCES. 

(, a ) To  show  that  when  three  concurrent  forces  are  in 
equilibrium,  each  is  equal  in  magnitude  and  opposite  in  di- 
rection to  the  resultant  of  the  other  two. 

{b)  To  show  that  any  one  of  these  forces  may  be 
replaced  by  a pair  of  rectangular  components. 

(D.  95,  96,  102-105.  G.  33-38.  K.  46-49.  W.  65-67,  71, 
72.) 

Apparatus. — Course  balance  and  set  of  weights,  3 uni- 
versal pulleys,  2 small  buckets,  can  of  shot,  4 iron  weights, 
4 strings  with  hooks  attached,  small  ring,  sheet  of  paper,  4 
thumb  tacks,  dividers,  and  block  of  wood,  vertical  board 
and  meter  stick. 

Theory  and  Method. — ( a ) Three  weights  are  allowed  to 
act  simultaneously  at  a point.  The  resultant  of  two  of 
these  forces  is  found  by  parallelogram  construction.  The 
parallelogram  law  states  that  when  two  concurrent  forces 
are  represented  in  direction  and  magnitude  by  the  direction 
and  length  of  two  straight  lines  drawn  from  one  and  the  same 
point , their  resultant  is  similarly  represented  by  that  diagonal 
of  the  parallelogram  constructed  on  these  two  lines  as  sides , 
which  starts  from  the  same  point.  The  resultant  so  found  is 
compared  in  direction  and  amount  with  the  third  force. 

(b)  One  of  the  forces  is  then  replaced  by  a vertical  and 
a horizontal  force,  which  act  at  the  same  point  as  the  original 
forces.  These  two  new  forces  are  adjusted  in  magnitude  un- 


38 


Manual  of  Experiments 


til  equilibrium  is  again  established  without  altering  the  two 
remaining  original  forces.  They  are  then  compared  with  the 
vertical  and  horizontal  components  of  the  third  force,  as 
found  by  the  parallelogram  construction. 

We  now  have  four  concurrent  forces  in  equilibrium,  but 
it  should  be  noticed  that  the  resultant  of  the  two  rectangular 
components  (the  original  force)  is  equal  and  opposite  to  the 
resultant  of  the  other  two  forces. 

Directions. — Find  the  mass  of  the  weights  on  the 
coarse  balance.  Clamp  a pulley  on  each  side  of  the  vertical 
board  and  fasten  a sheet  of  paper  to  the  latter  with  thumb 


Fig.  7 

tacks.  Suspend  the  smallest  and  one  of  the  larger  weights 
{A  & B)  over  the  pulleys  by  means  of  the  strings,  and  hook 
the  other  ends  of  the  latter  to  the  small  ring.  Suspend  a 
third  weight  from  this  ring  so  that  it  hangs  freely.  The 
pulleys  should  be  adjusted  so  that  they  turn  in  a plane 
parallel  to  the  board  and  so  that  the  weights  do  not  touch 
the  board  or  bench.  Adjust  the  height  of  the  pulleys  so 


In  General  Physics 


39 


that  the  center  of  the  forces  is  in  a convenient  position  for 
constructing  the  parallelogram  of  forces  on  the  paper. 
Displace  the  weights  slightly  and  allow  them  to  move 
freely  into  a position  of  equilibrium.  Then  draw  lines 
representing  the  forces  under  each  of  the  strings,  taking 
care  not  to  alter  the  position  of  the  latter  by  pressing  against 
them.  This  may  be  done  by  carefully  sliding  a small 
rectangular  block  of  wood  along  the  paper  until  one  face  is 
in  contact  with  the  string  near  one  end,  without  pressing 
against  it,  and  drawing  a short  mark  along  this  face  with  a 
sharp  pencil.  This  is  repeated  at  the  other  end  of  the  string 
and  a line  is  drawn  through  these  two  marks.  The  three 
lines  drawn  in  this  way  should  intersect  at  a point.  Begin- 
ning at  their  intersection  c,  lay  off  on  these  lines  to  some 
convenient  scale  (say  1 mm.  for  each  5 grams)  the  cor- 
responding forces.  On  the  upper  two  lengths,  ca  and  cd, 
as  sides,  construct  a parallelogram  abed.  The  resultant 
of  these  two  forces  is  the  diagonal  cb.  Measure  its  length, 
calculate  the  value  of  the  resultant  in  grams  weight  and 
compare  it  with  the  third  force.  Note  also  the  angle  be- 
tween be  and  the  vertical.  (cC  prolonged  upwards). 

In  the  same  way  find  the  resultant  of  A and  C and  com- 
pare it  with  B. 

It  should  be  noticed  in  this  construction  that  the  sides  of 
the  triangle  bed  are  parallel  to  and  proportional  to  the  three 
forces.  Whenever  three  concurrent  forces  are  in  equilibrium , 
they  are  proportional  to  the  sides  of  a triangle , the  sides  taken 
in  order  having  directions  parallel  to  the  forces. 

(b)  Lower  the  pulley  q until  its  upper  rim  is  in  the  hor- 
izontal line  ce,  and  replace  the  weight  B by  a bucket  partly 
filled  with  shot.  Clamp  a third  pulley  to  the  top  of  the 
board,  adjust  so  that  its  rim  is  vertically  above  c and  suspend 
a bucket  and  shot  over  it  by  means  of  a cord  hooked  to  the 


40 


Manual  of  Experiments 

ring.  Adjust  the  shot  in  these  two  buckets  until  the  ring 
returns  to  its  former  position  c , the  strings  be  and  ce  being 
respectively  vertical  and  horizontal.  Next  weigh  each 
bucket  with  its  shot.  Draw  the  horizontal  line  ce  and  drop 
the  vertical  de  from  the  end  of  the  line  representing  the 
force  B.  ce  and  ed  now  represent  respectively  the  horizontal 
and  vertical  components  of  B.  Measure  them,  calculate 
the  value  of  the  components  in  grams  weight  and  compare 
these  with  the  corresponding  weights  of  bucket  and  shot. 

Note: — For  accurate  results  the  parallelograms  must 
be  carefully  drawn  with  a sharp  pencil,  and  as  large  a scale 
as  possible  should  be  used  in  laying  off  the  forces. 

Question. — If  you  had  more  than  four  concurrent  forces 
in  a system,  how  would  you  show  that  they  were  in  equili- 
brium? 

4.  EQUILIBRIUM  OF  COPLANAR  CONCURRENT 
FORCES  BY  THE  TRIANGLE  OF  FORCES. 

To  verify  the  laws  for  three  forces  acting  upon  a point. 
(D.  52;  G.  38;  K.  48,  49;  W.  72.) 

Apparatus. — Force  table  and  accessories,  known  masses. 

* 

Description  and  Theory. — The  force  table  consists  of 
a cast  iron  disk  whose  edge  is  divided  into  degrees  and  which 
is  mounted  upon  a heavy  tripod  (See  Fig.  8).  Three  pul- 
leys, working  very  free  of  friction  in  cone  bearings,  can  be 
clamped  to  any  part  of  the  rim  of  the  table.  Known  masses 
are  fastened  to  light,  flexible  cords  which  pass  over  the  pul- 
leys and  are  attached  to  a small  ring  in  the  center  of  the 
table.  While  the  masses  are  being  adjusted  the  ring  is 
held  in  place  by  a pin.  If  suitable  masses  M,  N and  0 are 
placed  on  the  holders  at  the  ends  of  the  strings,  the  pulleys 


In  General  Physics 


41 


can  be  arranged  so  that  the  center  of  the  ring  remains  over 
the  center  of  the  table  when  the  peg  is  removed. 

If  a given  point,  say  the  center  ofthe  above  ring,  is  held 
in  equilibrium  by  three  forces,  the  condition  that  must  be 


satisfied  is,  that  the  forces  are  proportional  to  the  sides  of  a 
triangle,  the  sides  of  the  triangle  being  taken  in  the  order  of 
the  forces  and  having  directions  parallel  to  the  forces;  or,  in 
other  words,  the  sum  of  the  components  of  the  forces  resolved 
in  any  given  direction  must  be  zero. 

Suppose  for  equilibrium,  the  forces  take  up  positions  as 
shown  in  Fig.  9 a.  Let  the  forces  be  X = Mg,  Y = Ng  and 
Z = Og,  and  the  angles  opposite  the  forces  be  cc , 0, 
respectively.  The  condition  for  equilibrium  states  that 
the  forces  must  form  a triangle  when  lines  proportional 


42 


. Manual  of  Experiments 


to  the  forces  are  drawn  parallel  to  the  directions  in  which 
the  forces  are  acting.  If  the  position  of  the  forces  be  rep- 
resented by  Fig.  9 a , the  state  of  equilibrium  is  represented 
by  Fig.  9 b.  From  Fig.  9 b)  we  obtain,  by  trigonometry, 
the  relation, 


X Y Z 

sin  a sin  b sin  c 


or 


Mg  Ng  Og 
sin  a sin  b sin  c 


Since  the  acceleration  of  gravity  “ g”  enters  each  ratio, 
we  can  neglect  it  and  use  the  values  of  the  masses  instead 


Figs.  9a  and  9b 

of  the  forces,  they  being  proportional  to  each  other.  The 
angles  oc  f ft,  are  supplements  of  the  angles  a,  b,  c,  respec- 
tively, and  hence  the  final  relation  can  be  written 

M N 0 

sin  ft 


sin  cc 


sin  7 


(3) 


In  General  Physics 


43 


Directions. — See  that  the  force  table  is  level.  Place 
the  small  ring  in  the  center  of  the  table  and  put  the  pin 
through  it  to  hold  it  in  place.  Fasten  a flexible  cord  to 
each  of  the  three  weight  holders,  and,  having  run  the  strings 
over  the  pulleys,  fasten  them  by  means  of  the  small  hooks 
to  the  ring  in  the  center  of  the  table.  Add  masses  of  from 
300  to  600  grams  to  each  holder,  but  do  not  have  the  same 
mass  on  any  two  holders.  Place  one  of  the  pulleys  at  zero 
and  adjust  the  other  two  until  approximate  equilibrium 
is  obtained,  after  which  remove  the  pin  from  the  center 
of  the  ring.  Continue  to  adjust  the  pulleys  until  the  ring 
remains  over  the  center  of  the  table  when  the  latter  is  struck 
lightly  with  the  hand. 

When  equilibrium  is  obtained  and  the  center  of  the 
ring  is  over  the  center  of  the  table,  read  the  angles  between 

the  lines  of  action  of  the  forces,  or  the  angles  formed  by  the 

* 

strings  with  each  other,  and  designate  them  as  °c  y and  y 
(Fig.  9 a).  Determine  the  value  of  the  masses  at  the  end  of 
each  string  (known  masses  plus  the  holder)  and  test  the  re- 
lationship expressed  by  equation  (3). 

Repeat  the  determination  four  more  times  using  different 
combinations  of  the  masses  suspended  each  time.  Find  the 
mean  value  of  the  constant  (mass  divided  by  the  sine  of  the 
angle  opposite)  for  each  of  the  five  determinations.  For 
example, — 

X 

sin  cc 

Y 

First  Determination  — = 

sin 

Z 

sin  7 

Mean  = 


44 


Manual  of  Experiments 


Question. — (1)  What  is  the  maximum  percent  error 
from  the  mean,  (a)  in  your  best  determination,  (b)  in  your 
poorest? 

5.  EQUILIBRIUM  OF  COPLANAR  NONCONCUR- 
RENT FORCES. 

PARALLEL  & RANDOM  FORCES. 

V 

The  object  of  this  experiment  is  to  become  familiar  with 
the  laws  of  equilibrium 

(. a ) for  parallel  forces  acting  in  a plane. 

(b)  for  random  forces  acting  in  a plane.  (D.  81,  95-99, 
102-105.  G.  39,  40.  K.  53-63.  W.  68-73). 

1 

Apparatus. — 4 spring  balances,  3 steel  balls,  4 iron  pegs, 

2 boards,  pair  of  dividers,  4 iron  blocks,  meter  stick,  2 sheets 
of  paper.  { 

Theory  and  Method. — In  order  that  nonconcurrent 
forces  may  he  in  equilibrium  they  must  tend  to  produce  neither 
translation  nor  rotation.  Hence  there  are  two  rules  for  the 
equilibrium  of  such  forces.  We  are  going  to  consider  only 
forces  which  act  in  the  same  plane.  In  this  case — 

I If  the  forces  are  all  parallel , the  algebraic  sum  of  the 
forces  must  equal  zero , or  the  sum  of  the  forces  in  one  direction 
must  equal  the  sum  of  the  forces  in  the  opposite  direction. 

If  they  are  not  parallel  we  may  consider  their  components 
taken  parallel  to  two  rectangular  axes,  in  which  case  the 
sum  of  the  components  in  one  direction  must  equal  the  sum  of 
the  components  in  the  opposite  direction. 

II  In  either  case  the  algebraic  sum  of  the  moments  of  all 
the  forces  taken  about  any  point  in  the  plane  must  equal  zero 
in  order  that  there  may  be  no  rotation.  The  moment  of  a 
force  about  a point  is  a measure  of  the  turning  tendency  of 


In  General  Physics 


45 


the  force  about  an  axis  through  that  point.  It  is  numerically 
equal  to  the  product  of  the  force  into  the  perpendicular  distance 
from  the  point  to  the  line  of  direction  of  the  force.  A clockwise 
moment  is  one  which  tends  to  turn  about  the  point  in  the 
same  sense  as  the  hands  of  a clock.  An  anti-clockwise 
moment  tends  to  turn  in  the  opposite  direction..  For 
equilibrium  the  sum  of  the  clockwise  moments  must  equal  the 
sum  of  the  anti-clockwise  moments. 

Both  of  these  laws  must  be  fulfilled  by  every  system  of 
coplanar  forces  when  in  equilibrium.  We  may  find  that  a 
certain  system  does  not  produce  rotation  about  a certain 
point.  This  does  not  necessarily  mean  that  there  is  no 
rotation  about  some  other  point,  or  that  there  is  no  trans- 
lation, hence  we  cannot  say  that  the  system  is  in  equilibrium 
from  this  single  test,  but  in  general  must  apply  both  tests. 
Translation  may  be  considered  as  rotation  about  an  in- 
finitely distant  axis,  so  that  the  first  law  may  be  considered 
a special  case  of  the  second. 

The  apparatus  consists  of  a base  board  with  leveling 
screws,  and  a glass  plate  on  its  upper  side,  and  an  upper 
board  having  a series  of  equidistant  holes  in  parallel  rows. 
The  upper  board  rests  on  three  steel  balls  placed  on  the 
glass  plate.  The  object  of  this  arrangement  is  to  avoid 
friction  as  far  as  possible.  Small  spring  balances  with 
their  scales  graduated  in  ounces  are  held  by  pins  on  the  iron 
blocks  and  may  be  moved  to  any  position  relative  to  the 
board  and  to  each  other.  By  means  of  strings  they  may  be 
fastened  to  pegs  placed  in  any  of  the  holes  in  the  upper 
board  and  the  tension  on  them  may  be  adjusted  by  moving 
the  iron  blocks. 

The  spring  balance  consists  of  two  cylinders  one  of  which 
slides  within  the  other,  and  which  are  held  together  by  a 
spring.  The  extension  of  this  spring  is  by  Hooko’s  law  pro- 


46 


Manual  of  Experiments 


portional  to  the  weight  suspended  from  it,  and  a pointer  at- 
tached to  the  inner  cylinder  indicates  this  weight  directly  on 
the  graduated  scale  of  the  outer  cylinder.  In  a vertical  po- 
sition the  inner  cylinder  is  supported  by  the  spring,  whereas 
in  the  horizontal  position  it  is  not.  Therefore  the  no  load 
reading  will  be  different  in  the  two  cases,  and  this  difference 
must  be  added  to  the  zero  error  if  there  is  any,  when  the 
balance  is  used  in  the  horizontal  position. 

Directions. — ( a ) — Assume  the  balance  marked  A to  be 
correct  (without  zero  error)  and  find  the  zero  error  of  the 
other  balances  by  placing  each  on  an  iron  block,  joining  it  to 
A with  a string,  and  pulling  the  two  against  each  other  in  a 
straight  line  until  a force  of  one  or  two  pounds  is  indicated. 
Be  careful  in  using  the  balances  to  have  the  inner  cylinder 
in  line  with  the  outer  cylinder  to  avoid  errors  due  to  friction 
between  them.  Note  the  zero  error  of  the  balance  (the 
difference  between  the  reading  on  it  and  that  on  the  standard 
A),  and  mark  it  + or  — according  to  whether  it  is  to  be 
added  to  or  subtracted  from  future  readings  of  the  scale. 


Place  the  three  balls  between  the  glass  plate  and  the  upper 
board  (as  shown  in  figure  10)  and  adjust  the  leveling  screws 
until  the  board  stands  at  rest  without  touching  the  side 
stops.  Arrange  three  balances  as  shown,  fastening  the 
strings  to  pegs  which  may  be  placed  in  any  of  the  holes  of 


In  General  Physics 


47 


the  upper  board.  Adjust  the  positions  of  pegs  and  balances 
until  the  pull  on  the  single  balance  A is  as  large  as  possible 
and  the  strings  and  balances  are  in  line  and  in  the  same 
horizontal  plane,  the  strings  being  parallel  to  each  other 
as  shown  by  sighting  along  the  parallel  rows  of  holes.  The 
upper  board  should  rest  on  the  three  balls  and  be  free 
from  the  side  stops.  Note  the  pulls  on  the  balances,  correct 
them  for  the  zero  errors,  and  record  them.  Next  choose 
an  arbitrary  point  on  the  board  as  a center  and  measure 
the  perpendicular  distance  in  inches  from  this  point  to  the 
line  of  direction  of  each  force,  noting  at  the  same  time 
whether  the  force  tends  to  rotate  the  board  in  a clockwise  or 
an  anticlockwise  direction  about  the  arbitrary  center. 
Then  calculate  the  moment  of  each  force  and  record  it. 

On  the  data  sheet  compare  the  sum  of  the  forces  pulling 
in  one  direction,  say  east,  with  the  sum  of  the  forces  pulling 
in  the  opposite  direction,  and  the  sum  of  the  clockwise  mo- 
ments with  that  of  the  anticlockwise  moments,  giving  the 
percent  difference  between  them  (calculated  from  their 
mean)  in  each  case. 

(b)  Place  a sheet  of  paper  on  the  top  board  and  arrange 
four  balances  so  that  the  forces  acting  on  the  board  are  not 
parallel,  but  are  placed  at  random,  the  iron  pegs  being 
punched  through  the  paper.  When  equilibrium  has  been 
established  (the  balances  being  in  line  with  their  strings  and 
the  top  board  free  from  sidestops),  rule  a line  under  each 
string  as  in  Exp.  3,  and  record  the  correct  values  of  the 
forces  pulling  on  the  balances  beside  each  line.  Remove 
the  pegs  and  paper  and  draw  a pair  of  rectangular  axes, 
say  east  and  west,  north  and  south.  Lay  off  on  each  of  the 
four  lines  to  some  convenient  scale,  the  corresponding 
forces,  and  from  the  ends  of  the  lines  so  laid  off  draw  the 
components  of  the  forces  parallel  to  the  two  axes.  Record 


48 


Manual  of  Experiments 


the  values  of  the  components  and  their  direction.  (E.  N., 
etc.,  for  east,  north,  etc.) 

Choose  an  arbitrary  point  p on  the  paper  as  a center  and 


measure  the  perpendicular  distance  in  inches  from  this  point 
to  each  of  the  force  lines  (extended  if  necessary  as  in  Fig. 
11).  The  arbitrary  center  should  be  so  chosen  that  the 
perpendicular  distances  are  as  large  as  possible,  to  reduce 
errors  in  measurements,  and  the  distances  should  be  es- 
timated to  1 64  of  an  inch.  Note  which  forces  give  clock- 
wise and  which  anticlockwise  moments  and  record  them 
properly. 

On  the  data  sheet  compare  the  sum  of  the  east  compon- 
ents with  the  sum  of  the  west  components;  the  sum  of  the 
north  with  the  sum  of  the  south  components;  and  the  sum 
of  the  clockwise  with  the  sum  of  the  anticlockwise  moments, 
and  calculate  the  precent  difference  between  them  in  each 
case. 

Questions — 1 — A beam  10  feet  long  weighing  200  lbs., 
along  which  a pulley  and  chain  travels,  is  supported  at  one 


In  General  Physics 


49 


end  by  a wall,  at  the  other  by  a column  which  cannot  support 
more  than  600  lbs.  If  the  pulley  with  its  load  weighs  800 
lbs.  how  close  can  it  be  drawn  to  the  column  without  over- 
loading the  latter.  Apply  both  laws  of  equilibrium. 

2. — In  performing  this  experiment  with  4 random  forces, 
it  was  found  that  two  of  them  intersected  in  a point  0; 
namely  a force  A of  10  oz.  directed  30°  W.  of  N.,  and  B a 
force  of  20  oz,  directed  60°  E.  of  N.  A third  force  C of 
30  oz.  directed  45°  E.  of  S.  had  a moment  arm  3 inches  long 
in  a direction  S.  W.  from  O.  Find  the  direction  and  magni- 
tude of  the  fourth  force  and  its  distance  and  direction  from  O. 


6.  PARALLEL  FORCES. 


{a)  To  verify  the  law  for  the  equilibrium  of  moments. 
(, b ) To  determine  the  value  of  an  unknown  mass,  (c)  To 
find  the  mass  of  a meter  stick.  (D.  97-100;  G.  39-41.  K. 
59-64;  W.  69,  70.) 

Apparatus. — Clamp  stand,  meter  stick,  set  of  known 
masses,  three  lever  holders,  coarse  balance,  unknown  mass. 

Description  and  Theory. — Moment  of  a force  is  the 
term  used  to  express  the  value  of  a force  times  its  arm, 
where  the  arm  is  the  distance  measured  at  a right  angle  from 
the  line  of  action  of  the  force  to  the  axis  about  which  rotation 
is  capable  of  taking  place.  A body,  free  to  rotate  about  an 
axis,  is  in  equilibrium  only  when  the  two  following  conditions 
are  satisfied:  first,  when  the  sum  of  the  components  of  the 
forces  in  any  direction  is  equal  to  the  sum  of  the  components 
in  the  opposite  direction;  and  second,  when  the  sum  of  the 
moments  of  the  forces  tending  to  produce  rotation  in  one  di- 
rection is  equal  to  the  sum  of  the  moments  of  the  forces  tend- 
ing to  produce  rotation  in  the  opposite  direction.  The  latter 


50 


Manual  of  Experiments 


condition,  or  law,  may  be  verified  by  a simple  arrangement 
consisting  of  a meter  stick  supported  on  a knife-edge  at  its 
center  of  gravity  and  having  known  masses  supported  on 
knife  edges  at  such  distances  from  the  center  of  gravity  that 
the  product  of  one  known  mass  plus  the  mass  of  the  support- 
ing knife  edge  times  their  distance  from  the  center  of  gravity 
is  equal  to  the  corresponding  product  for  the  other  mass. 

The  law  can  be  applied  to  determine  an  unknown  mass  by 
replacing  one  of  the  known  masses  by  the  unknown.  The 
product  of  the  unknown  mass  times  its  distance  from  the  cen- 
ter of  gravity  of  the  supporting  meter  stick  is  equal  to  the 
product  of  the  known  mass  times  its  distance  from  the  cen- 
ter of  gravity.  In  all  cases,  the  masses  of  the  supporting 
knife  edges  which  is  stamped  on  the  holder  must  be  taken 
into  account. 

The  mass  of  a meter  stick  can  be  determined  by  first 
finding  the  center  of  gravity  of  the  meter  stick  by  balancing 
it  alone  and  then,  having  placed  the  center  of  gravity  a short 
distance  from  the  supporting  knife  edge,  putting  a known 
mass  at  such  a distance  from  the  supporting  knife  edge  as  to 
obtain  equilibrium.  The  product  of  the  mass  of  the  meter 
stick  times  the  distance  of  the  center  of  gravity  of  the  stick 
from  the  supporting  knife  edge  must  equal  the  product  of  the 
known  mass  times  its  distance  from  the  point  of  support. 

Directions. — To  verify  the  law  for  the  equilibrium  of 
moments,  support  the  meter  stick  by  one  of  the  lever  holders 
and  adjust  the  same  till  equilibrium  is  obtained.  Put  a 
lever  holder  on  each  end  of  the  meter  stick  and  hang  masses 
A and  B (Fig.  12),  of  200  and  300  grams  on  the  holders. 
Adjust  the  position  of  the  holders  (at  least  20  cms.  from  the 
supporting  knife  edge)  with  the  suspended  masses  till 
equilibrium  is  obtained.  Note  the  distance  from  the 
supporting  knife  edge  to  the  knife  edge  of  each  of  the  holders. 


In  General  Physics 


51 


Test  the  relation  for  the  equilibrium  of  moments  by  mul- 
tiplying the  mass  of  the  holder  and  the  suspended  weight 
by  its  distance  from  the  supporting  knife  edge.  The 
product  of  one  suspended  mass  plus  the  mass  of  the  holder 


Fig.  12 


supporting  it  times  their  distance  from  the  supporting 
knife  edge  should  be  equal  to  the  corresponding  product 
for  the  other  mass.  Make  a second  adjustment  of  the  masses 
A and  B for  different  distances  from  the  supporting  knife 
edge  and  test  the  relation  for  the  equilibrium  of  moments. 
Use  another  combination  of  masses  300  and  400  grams,  and 
obtain  the  products  of  mass  times  the  distance  from  the 
point  of  support  for  two  different  positions  of  the  masses 
when  in  equilibrium. 

In  order  to  determine  the  value  of  an  unknown  mass  it 
is  only  necessary  to  substitute  the  unknown  mass  X for 
either  of  the  masses  A or  B above,  and  find  the  position 
for  equilibrium.  The  unknown  mass  X is  determined  from 
the  fact  that  the  product  of  X times  its  distance  from  the 
point  of  support  must  be  equal  to  the  product  of  the  known 
mass  times  its  distance  from  the  supporting  knife  edge. 
Make  three  different  determinations  of  the  value  of  X and 
find  their  mean.  Remember  that  the  masses  of  the  knife 
edges  supporting  the  known  and  . unknown  masses  must 
always  be  considered. 


52 


Manual  of  Experiments 

The  mass  of  the  meter  stick  is  found  as  follows:  Find 

the  position  of  the  center  of  gravity  of  the  meter  stick  by 
suspending  it  from  one  of  the  knife  edges.  Move  the  stick 
so  that  its  center  of  gravity  is  from  15  to  20  centimeters  from 
the  supporting  knife  edge  and  suspend  a 100  gram  mass  by 
means  of  a lever  holder  at  such  a position  as  to  produce 
equilibrium.  Note  the  distances  of  the  100  gr.  mass  and 
the  center  of  gravity  from  the  supporting  knife  edge.  Then, 
if  M is  the  mass  of  the  meter  stick,  M times  the  distance  of 
the  center  of  gravity  of  the  stick  from  the  point  of  suspension 
is  equal  to  100  grams  plus  the  mass  of  the  holder  supporting 
it  times  its  distance  from  the  knife  edge  of  the  holder  sup- 
porting the  system. 

Make  two  more  adjustments  for  equilibrium  using  dif- 
ferent distances  and  determine  the  mass  of  the  meter  stick 
in  each  case.  Calculate  the  mean  of  the  three  determina- 
tions of  the  mass  of  the  meter  stick. 

Question. — What  two  conditions  must  be  satisfied  for 
equilibrium  of  non-concurrent  parallel  forces?  Show  how, 
if  the  sum  of  all  the  forces  downward  did  not  equal  the  sum 
of  the  forces  upward,  the  other  condition  could  be  fulfilled 
and  still  not  haye  the  bar  in  equilibrium.  What  kind  of 
motion  would  we  then  obtain? 

7.  THE  FINE  BALANCE. 

In  this  experiment  we  are  to  become  familiar  with  the 
vibration  method  of  using  the  fine  balance.  (D.  135;  G.  73- 
76:  K.  69,  70.  W.  95.) 

Apparatus. — Set  of  weights  (2mg.  to  100  g.,)  lead  plate 
to  be  weighed,  spirit  level,  and  fine  balance. 

Description  and  Method. — The  fine  balance  is  to  be 


In  General  Physics 


53 


used  whenever  an  object  is  to  be  weighed  to  less  than  0.1 
gram.  It  consists  of  a rigid  beam  with  a central  knife  edge 
resting  on  a polished  agate  plate  at  the  top  of  a pillar, 
and  two  equidistant  knife  edges  KK  from  which  the  balance 
pans  are  suspended  by  means  of  stirrups  with  polished  agate 
plates.  (See  Fig.  13.)  Agate  is  used  in  order  to  make  the 
friction  in  swinging  very  slight.  To  protect  the  balance 
from  disturbing  air  currents  and  dust,  it  is  enclosed  in  a glass 
case  having  a heavy  glass  base  and  supported  on  leveling 
screws.  To  avoid  jarring  and  chipping  of  the  knife  edges 
when  the  balance  is  not  in  use  or  while  ‘ ‘ weights  ’ ’ or  objects 
are  being  added  to  or  taken  from  the^pan,  an  arrestment 


■ 


Fig.  3 


54 


Manual  of  Experiments 


device  is  used.  This  is  operated  a thumb  nut  (N)  at 
base  of  balance  which,  on  turning,  lifts  the  beam  off  the  agate 
plate  and  the  stirrups  off  the  knife  edges.  Two  stops  rest 
against  the  pan  to  prevent  them  from  swinging.  To  release 
them  a button  ( B ) is  pushed  in  until  it  is  caught  by  the 
lever  ( A ).  A long  pointer  ( P ) attached  to  the  beam  in- 
dicates the  position  of  rest  on  a scale  at  the  base  of  the 
pillar. 

As  the  milligram  weights  are  quite  small  and  incon- 
venient to  handle,  a rider  ( R ) is  often  used  in  place  of  weights 
less  than  10  mg.  It  is  a piece  of  wire  which  weighs  exactly 
6 mg.  and  is  so  shaped  that  it  may  be  placed  astride  of  a 
scale  on  the  beam.  By  means  of  a system  of  levers  operated  . 
from  the  outside  by  a thumb  nut  (L)  at  the  right  of  the  case, 
this  rider  may  be  moved  to  any  point  on  the  scale.  When 
just  above  one  of  the  end  knife  edges  its  effect  is  the  same 
as  if  it  were  placed  in  the  pan.  When  placed  at  one  sixth 
this  distance  from  the  central  knife  edge  its  moment  about 
the  fulcrum  will  equal  that  of  one  sixth  of  its  mass,  or 
(1  mg.),  placed  in  the  pan.  The  marked  divisions  on  the 
scale  represent  milligrams,  the  smallest  divisions  tenths  of  a 
milligram. 

Adjustable  nuts  working  on  horizontal  screws  at  the 
ends  of  the  beam,  make  it  possible  to  balance  the  latter 
alone. 

A good  balance  should  fulfill  the  following  conditions: 
(1)  The  balance  must  be  true,  that  is,  the  beam  must  re- 
main horizontal  whenever  equal  masses  are  placed  in  the 
scale  pans.  (2)  It  must  be  sensitive,  i.  e.  small  differences 
in  the  masses  in  the  two  pans  should  cause  an  appreciable 
deviation  of  the  beam  from  its  horizontal  position.  (3)  It 
must  be  stable;  that  is,  after  the  beam  has  been  disturbed 
from  its  equilibrium  position,  it  must  return  to  it.  (4)  It 


In  General  Physics 


55 


should  have  a short  period;  i.  e.  when  the  beam  has  been  dis- 
turbed and  oscillates  before  coming  to  rest,  the  time  re- 
quired for  it  to  make  a complete  swing  should  be  short. 

In  order  that  the  balance  may  be  true,  the  beam  should 
be  inflexible  and  its  arms  of  equal  length.  To  be  sensitive 
the  knife  edges  should  be  parallel  and  in  the  same  plane,  the 
arms  should  be  long,  and  the  center  of  gravity  of  the  beams 
should  be  just  below  the  central  knife  edge.  In  order  that 
the  beam  may  quickly  return  to  its  equilibrium  position  when 
displaced,  the  arms  should  be  short  and  the  distance  between 
center  of  gravity  of  beam  and  knife  edge  large.  These 
conditions  conflict  with  those  for  sensitiveness  and  in 
• practical  balances  a compromise  between  them  must  be 
chosen. 

The  approximate  weight  of  an  object  may  be  obtained 
by  the  equal  swing  method.  Small  bits  of  paper  are  added 
to  the  pans  until  the  pointer  swings  equidistant  on  either 
side  of  the  center  division.  Then  the  object  is  placed  in  the 
center  of  one  pan  and  weights  are  added  to  the  other  until  the 
pointer  swings  as  nearly  as  possible  as  before,  equidistant 
on  either  side  of  the  center  division.  Then  the  object  and 
known  weights  will  have  approximately  the  same  mass.  To 
find  the  mass  more  accurately,  it  would  be  necessary  to  find 
the  true  resting  point  (i.  e.  the  point  on  the  scale  at  which 
the  pointer  would  finally  come  to  rest  if  allowed  to  swing 
freely).  As  the  period  of  the  balance  (the  time  required 
for  a complete  vibration  of  the  beam)  is  usually  long,  it  is 
preferable  to  use  the  following  “vibration  method  of  weigh- 
ing”. The  resting  point  with  no  load  in  the  pans  is  found 
by  observing  an  odd  number,  say  five,  consecutive  turning 
points  of  the  pointer,  estimating  to  tenths  of  a scale  division. 
The  reason  for  taking  an  odd  number  of  turning  points  is 
the  following: — The  amplitude  of  swings  to  either  side  of 


56 


Manual  of  Experiments 


the  resting  point  is  continually  decreasing  by  approximately 
equal  amounts,  say  a divisions  for  a few  equal  swings  (see 
Fig.  14).  The  mean  of  the  first  and  second  turning  points 
would  be  approximately  a/ 2 divisions  to  one  side  of  the  rest- 
ing point.  The  mean  of  the  second  and  third  turning  point 
would  be  about  a/2  divisions  to  the  other  side  of  the  resting 
point.  The  mean  of  two  means  would  be  very  near  the 
resting  point.  If  five  turning  points  were  observed  instead 
of  three,  the  mean  of  the  means  of  the  five  readings,  taken 
two  and  two  consecutively,  would  be  still  nearer  the  point, 
whereas  if  any  even  number  of  points  or  an  odd  number  of 
swings  were  observed  the  mean  would  always  be  to  one  side 
or  the  other  of  the  true  resting  point.  Therefore,  to  find 
the  true  resting  point,  average  the  mean  of  n (odd)  turning 


points  on  one  side  {a,  c,  e,  Fig.  14.)  with  the  mean  of  (n  — 1) 
intervening  turning  points  on  the  other  (b,  d.  Fig.  14).  As 
an  example,  if  the  pointer  swings  from  6.0  to  12.8  and  back 
to  6.8,  the  two  means  would  be  9.4  and  9.8  respectively 
and  the  mean  of  the  two  means  of  resting  point  would  be 
9.6.  This  no  load  resting  point  we  will  call  S0.  If  now  the 
object  to  be  weighed  is  placed  in  one  pan  and  known  weights 
in  the  other  until  they  balance,  the  load  resting  point 
Si  may  be  found  in  the  same  way  as  SQ  and  in  general  will 


In  General  Physics 


57 


be  different  from  S0.  But  it  is  evident  that  the  masses  in 
the  pans  can  only  be  equal  when  the  pointer  comes  to  rest 
at  S0?  since  this  is  its  equilibrium  position.  We  must  there- 
fore determine  what  weight  must  be  added  to  or  taken  from 
the  known  weights  in  the  pan  in  order  to  make  the  pointer 
come  to  rest  at  SQ  (instead  of  Si)  when  the  balance  is  loaded. 
To  do  this  we  add  a small  weight,  say  5 mg.  to  the  known 
weights  and  observe  the  new  load  resting  point  S2.  Since 
0.005  grams  moves  the  resting  point  from  Si  to  S2  the  number 
of  grams  n necessary  to  move  one  division  will  be  0.005  -f- 
(S2  — Si).  The  reciprocal  of  n is  called  the  sensibility  of  the 
balance;  i.  e.  the  number  of  divisions  through  which  the 
pointer  is  moved  by  1 mg.  in  the  pan.  In  order  that  the 
load  resting  point  may  be  changed  from  Si  to  SQ  we  must 
change  the  known  weights  by  n (Si  — S0)  g.  If  the  known 
weights  are  in  the  right  pan  we  must  add  this  amount  to 
them  when  Si  is  greater  than  SG  and  subtract  it  when  Si 
is  less  than  Sc,  in  order  to  obtain  the  true  mass  of  the  object. 

Directions. — Caution.  Always  keep  the  case  closed  ex- 
cept when  it  is  necessary  to  adjust  the  weights.  Lower  the 
arrestment  slowly  and  carefully,  avoiding  so  far  as  possible, 
all  jarring  of  the  balance.  Always  raise  the  arrestment  be- 
fore changing  the  weights  in  the  pans.  Handle  the  weights 
with  tweezers  to  avoid  corroding  them  or  changing  their 
weight.  Do  not  attempt  to  make  any  of  the  finer  adjust- 
ments with  the  screws  on  the  balance.  If  such  adjustments 
are  necessary  call  on  one  of  the  instructors  to  make  them. 

Adjust  the  leveling  screws  until  the  base  of  the  balance 
is  horizontal  as  shown  by  the  spirit  level.  Lower  the  arrest- 
ment until  the  pointer  swings  freely  and  note  whether  the 
no  load  resting  point  is  within  one  or  two  divisions  of  the 
center.  If  it  is  not,  drop  small  bits  of  paper  into  the  proper 
pan  until  it  is  within  this  distance  from  the  center.  Then 


58 


Manual  of  Experiments 

find  the  no  load  point  Sc  accurately,  by  taking  five  con- 
secutive turning  points,  (estimating  to  1/10  of  a scale 
division)  and  averaging  the  mean  of  those  on  one  side  with 
the  mean  of  those  on  the  other  side.  Next  raise  the  arrest- 
ment and  place  the  object  to  be  weighed  in  one  pan  and 
known  weights  in  the  other  to  balance.  Lower  the  arrest- 
ment a trifle  and  note  which  way  the  beam  tilts  {i.  e.  which 
of  the  pans  has  the  heavier  mass  in  it) ; then  raise  the  arrest- 
ment and  adjust  the  weights  to  secure  a better  equilibrium. 
Repeat  until  there  is  very  little  tendency  to  tilt  the  beam. 
Time  may  be  saved  by  trying  the  weights  in  the  order  in 
which  they  come  in  the  box,  and  replacing  them  in  similar 
order.  The  finer  adjustments  may  be  made  with  the 
rider.  Record  the  mass  m of  known  weights. 

Determine  the  load  resting  point  Si  from  five  con- 
secutive turning  points.  Add  5mg  or  less  to  the  known 
weights,  determine  the  load  resting  point  S2,  and  calculate 
n,  the  reciprocal  of  the  sensibility  of  the  balance.  Remove 
the  weights  and  object  from  the  pans  (while  the  arrestment 
is  raised,  of  course)  and  determine  the  no  load  resting  point 
S3.  If  the  latter  differs  by  less  than  one  division  from  first 
value  S0  the  weighing  is  sufficiently  accurate.  If  the 
difference  is  greater  than  one  division  the  experiment  should 
be  repeated. 

From  the  data  taken,  calculate  the  true  mass  M of  the 
object.  M = m =±=  n (Si  — SQ) 

Question. — If  the  center  of  gravity  of  the  beam  could 
be  raised  or  lowered  by  means  of  an  adjustable  nut,  how 
would  the  balance  be  affected  in  each  case. 

' - 

* 8.  THE  PENDULUM. 

(a)  The  laws  of  the  simple  pendulum. 

(b)  The  determination  of  the  acceleration  of  gravity 


In  General  Physics 


59 


with  the  Kater’s  pendulum.  (D.  117,  120,  G.  56,  81-84, 
K.  127,  128,  141,  W.  112-118.) 

Apparatus. — Two  lead  balls,  fine  thread,  vernier  calipers, 
reversible  pendulum  and  support;  beam  compass,  meter 
stick,  and  seconds  clock. 

Description,  Method  and  Theory. — (a)  If  it  were 
possible  to  have  a material  particle  concentrated  in  a point 
and  suspended  by  a weightless,  inextensible  thread,  this 
would  be  a theoretical  simple  pendulum.  These  conditions 
can  not  be  fulfilled  in  practice,  and  the  practical  simple 
pendulum  consists  of  a small  heavy  bob  (usually  spherical) 
suspended  by  a light  and  practically  inextensible  thread  or 
wire.  Fig.  15.  The  period  of  such  a pendulum  is  the 
time  required  for  a complete  oscillation,  or  double  swing; 
that  is,  the  time  which  elapses  between  two  successive  pas- 
sages of  the  pendulum  in  the  same  direction , through  some 
- point  of  its  path,  such  as  the  center  of  its  swing.  The  length 
of  “the  equivalent  simple  pendulum ” (that  is,  the  length  of 
the  theoretical  simple  pendulum  which  has  the  same  period) 
is,  for  a spherical  bob  (neglecting  the  mass  of  suspension), 

/ = d + ( 2r2/5d ) 

where  d is  the  distance  from  the  point  of  support  to  the 
center  of  the  bob,  and  r is  the  radius  of  the  bob.  For 
practical  purposes  the  length  of  the  pendulum  is  taken  as 
equal  to  d.  The  period  of  such  pendulum  is 

T = 2w  V //  g, 

where  g is  the  acceleration  of  gravity.  The  period  is  inde- 
pendent of  the  mass  of  the  bob  and  of  the  arc  of  its  swing, 
provided  that  arc  does  not  exceed  about  4 degrees. 

We  wish  to  test  the  variation  of  the  period  of  such  a pen- 
dulum with  its  length  and  with  its  mass.  In  order  to  find 


60 


Manual  of  Experiments 


the  variation  of  any  physical  quantity  with  one  of  its  factors, 
it  is  evident  that  all  other  factors,  must  be  kept  constant. 
We  therefore  take  two  pendulums  of  equal  mass  but  of 
different  lengths,  in  order  to  compare  the  variation  of  their 
periods  with  their  lengths,  and  two  pendulums  of  equal 
lengths  but  of  different  masses,  to  note  a possible  variation 
of  period  with  mass.  If  the  student  is  working  alone  he 
may  determine  the  period  of  the  pendulum  with  a stopwatch, 


by  starting  and  stopping  the  watch  at  the  beginning  and 
end,  respectively,  of  a definite  number  of  oscillations.  If 
two  students  work  together,  an  ordinary  watch  with  a 
seconds  hand  may  be  used,  one  student  observing  the  number 
of  oscillations  of  the  pendulum,  the  other  the  number  of 
seconds  which  elapse.  The  first  student  should  call  ‘ ‘ ready  ’ ’ 
a few  seconds  before  he  begins  to  count  the  oscillations. 
Then  at  the  exact  instant  when  the  pendulum  passes  through 
its  center  of  path  (position  of  rest)  he  should  call  11  start  ”, 
beginning  at  the  same  time  to  count  the  passages  of  the  pen- 
dulum through  this  point  in  the  same  direction , starting  with 


In  General  Physics 


61 


zero  for  its  first  passage.  The  other  student  should  note 
the  exact  time  at  this  instants  On  the  95th  or  96th  passage 
the  first  student  again  calls  “ready”  to  warn  the  second 
student,  and  on  the  instant  that  the  pendulum  completes 
its  100th  passage  through  the  position  of  rest  he  calls  “stop  ’ 
upon  which  the  second  student  again  notes  the  exact  time. 
If  the  students  have  no  watch  at  hand,  they  may  observe 
the  time  on  the  seconds  clock  in  the  laboratory.  This 
clock  has  a pendulum  which  completes  one  swing  (half 
oscillation)  every  second.  It  has  a device  for  closing  an 
electric  circuit  every  time  it  passes  through  its  position  of 
rest.  In  this  circuit  are  included  a galvanic  cell  and  a tele- 
graphic sounder,  and  every  time  the  circuit  is  closed,  the 
sounder  gives  a loud  click,  thus  indicating  seconds.  When 
all  is  ready  for  the  trial  one  of  the  students  calls  “ready”. 
The  second  student  then  observes  both  the  passage  of  the 
pendulum  through  its  position  of  rest  and  the  click  of  the 
sounder.  At  the  instant  when  both  events  occur  exactly  in 
unison  he  calls  “start  ’ ’,  beginning  at  the  same  time  to  count 
the  clicks  of  the  sounder,  zero,  one,  two,  three,  etc.,  while 
the  first  student  counts  the  passages  of  the  pendulum  in  the 
same  direction,  beginning  zero,  one,  two,  etc.  When  nearly 
one  hunderd  swings  have  been  completed  the  first  student 
begins  to  observe  the  clicks  of  the  sounder  as  well  as  the 
passages  of  the  pendulum  and  when  he  sees  that  the  two 
nearly  coincide  he  calls  “ready”,  and  at  the  exact  instant 
when  both  occur  simultaneously  he  calls  “stop”,  noting  the 
exact  number  of  passages,  while  the  second  student  notes 
the  exact  number  of  clicks  up  to  this  instant.  To  avoid 
the  tedious  counting  of  oscillations  we  may  proceed  as 
follows : — 

Note  as  accurately  as  possible  with  a watch  or  with  the 
seconds  clock  the  time  of  10  complete  oscillations,  and  calcu- 


62 


Manual  of  Experiments 


late  the  approximate  period.  Then  note  the  exact  instant 
of  the  transit  of  the  pendulum  through  its  position  of  rest. 
Calculate  the  approximate  time  required  for  100  oscillations, 
and  at  the  expiration  of  this  time  again  note  the  exact  instant 
of  transit  in  the  same  direction  as  before.  Divide  the  inter- 
val of  time  between  these  twro  transits  by  the  approximate 
period.  The  whole  number  coming  nearest  to  this  quotient 
will  be  the  exact  number  of  oscillations  in  this  interval  and 
on  dividing  the  same  interval  by  the  exact  number  of  oscil- 
lations we  get  the  exact  period. 

The  length  of  the  pendulum  is  measured  with  a beam  com- 
pass which  consists  of  a straight  wooden  beam  along  which 
two  trammel  points  may  be  moved  and  clamped  at  any  de- 
sired position.  The  distance  between  the  trammel  points 
may  be  measured  with  a meter  stick. 

The  average  time  of  one  complete  oscillation  should  be 
calculated  and  a comparison  made  between  the  periods  for 
different  lengths  and  for  different  masses. 

(b)  The  compound  pendulum  differs  from  the  simple 
pendulum  in  that  its  mass  is  not  concentrated  at  one  point. 
Evidently  every  physical  pendulum  is  really  a compound 
pendulum.  That  point  in  the  pendulum  where  the  particles 
are  oscillating  with  the  same  period  that  they  would  have  if 
they  were  bobs  of  simple  pendulums  suspended  from  the  same 
point  of  support  as  the  actual  pendulum,  is  called  the  center 
of  oscillation,  the  point  of  support  being  called  the  center  of 
suspension.  These  two  points  are  interchangeable.  That 
is,  if  the  pendulum  is  suspended  from  the  center  of  oscillation, 
the  previous  center  of  suspension  becomes  the  center  of  os- 
cillation. If  the  whole  mass  of  the  pendulum  were  concen- 
trated at  the  center  of  oscillation,  the  period  of  the  resulting 
simple  pendulum  would  be  the  same  as  that  of  the  actual 
pendulum.  Therefore  the  length  of  “the  equivalent  sim- 


In  General  Physics 


63 


pie  pendulum  ’ ’ is  the  distance  between  the  center  of  oscilla- 
tion and  the  center  of  suspension.  The  fact  that  these  two 
centers  are  interchangeable  makes  it  possible  to  determine 
their  relative  positions  accurately  and  so  makes  possible  the 
accurate  determination  of  the  length  of  the  equivalent  simple 
pendulum.  The  period  may  also  be  determined  accurately 


Fig.  16 

i 

L 

and  consequently  an  exact  determination  of  the  acceleration 
of  gravity  is  possible  from  the  relation 

£ = 4t t2//P 


64 


Manual  of  Experiments 


(see  expression  for  period  of  simple  pendulum). 

For  this  purpose  a Kater’s  reversible  pendulum  is  used. 
It  consists  of  a metal  rod,  carrying  two  adjustable  steel  knife 
edges  KK  with  their  edges  turned  toward  each  other.  See 
Fig.  16.  At  one  end  a heavy  mass  A is  fixed,  while  a light 
mass  B with  adjusting  nuts  may  be  clamped  at  any  position 
along  the  length  of  the  pendulum.  The  knife  edges  are  fixed 
at  some  definite  distance  apart  and  the  pendulum  is  swung 
first  from  one  knife  edge,  then  from  the  other,  its  period 
being  determined  each  time  and  the  weights  adjusted  until 
the  periods  are  exactly  equal.  The  period  and  the  length 
between  the  knife  edges  may  now  be  accurately  measured 
and  the  acceleration  of  gravity  calculated. 

Directions. — ( a ) Relation  between  period  T and  length 
h — -Fasten  the  smaller  ball  to  the  thread  and  suspend  it  from 
the  support.  Adjust  the  length  of  the  thread  until  it  is 
about  60  or  64  cm.  Adjust  the  pointer  on  the  support  until 
it  is  just  under  the  center  of  the  bob  when  the  latter  hangs  at 
rest.  Place  the  beam  compass  in  line  with  the  pendulum, 
one  end  resting  on  the  floor,  and  adjust  the  trammel  points 
until  they  stand  exactly  at  the  ends  of  the  thread.  Lay  the 
beam  compass  on  the  table  and  measure  the  distance  between 
the  trammel  points  by  means  of  a meter  stick  and  record  it 
on  the  data  sheet.  Measure  the  diameter  of  the  bob  with 
the  vernier  calipers  in  four  or  five  different  positions,  and  re- 
cord the  mean  diameter.  Then  calculate  the  length  of  the 
pendulum. 

Set  the  pendulum  to  swinging  through  a small  arc.  Be 
careful  to  have  it  swinging  in  a plane;  that  is,  so  that  the 
path  of  the  bob  is  a straight  line,  not  an  ellipse.  Find  the 
time  for  one  complete  vibration  by  the  method  explained 
above,  noting  the  passage  of  pendulum  past  the  pointer. 
Repeat  the  trials  until  three  values  of  the  period  are  obtained 


In  General  Physics 


65 


which  do  not  differ  by  more  than  0.01  second  and  then 
record  the  mean  of  these  trials. 

Repeat  these  measurements  with  a pendulum  about  one 
meter  long. 

Show  by  your  data  that  the  period  is  directly  proportional 
to  the  square  root  of  the  length,  by  calculating  the  ratio  be- 
tween them  in  each  case.  Record  these  ratios  and  the  per- 
cent difference  between  them  calculated  from  their  mean. 

Independence  of  Period  and  Mass. — Fasten  the  large  lead 
ball  to  the  thread  and  suspend  it  as  before.  Keeping  the  po- 
sition of  the  trammel  points  on  the  beam  compass  exactly  as 
they  were  for  the  last  measurement  of  length,  adjust  the 
length  of  the  pendulum  until  the  center  of  the  bob  is  at  the 
lower  trammel  point.  The  pendulum  then  has  very  nearly 
the  same  length  as  the  previous  pendulum  with  the  smaller 
bob.  Proceed  to  find  the  period  of  the  pendulum  and  com- 
pare it  with  that  of  the  previous  pendulum  of  the  same 
length.  Record  these  periods  and  the  percent  difference 
between  them  calculated  from  their  mean. 

(, b ).  The  determination  of  g. — Using  the  Kater’s  pen- 
dulum let  one  knife  edge  and  the  two  weights  remain  fixed, 
and  the  second  knife  edge  K movable.  Determine  the 
period  of  the  pendulum  about  both  knife  edges  for  one 
position  of  K,  and  measure  the  distance  ill,J  between  knife 
edges.  Then  shift  K about  5 cm.  and  repeat  the  measure- 
ments. Repeat  until  you  have  four  periods  about  each 
knife  edge  and  the  corresponding  distances  between  them. 
Plot  two  curves  having  values  of  / as  abscissae  and  correspond- 
ing periods  about  both  knife  edges  as  ordinates.  These 
two  curves  will  intersect  and  the  ordinates  of  the  point  of 
intersection  will  be  the  period  and  length  of  the  equivalent 
simple  pendulum  respectively. 

For  accurate  results  the  distance  between  knife  edges 


66 


Manual  of  Experiments 


should  be  measured  with  a comparator,  but  for  our  purposes 
we  may  use  the  length  measured  with  the  beam  compass  and 
meter  stick. 

From  the  period  and  length  of  the  equivalent  simple 
pendulum  calculate  the  value  of  g and  record  it  on  the  data 
sheet. 

Question. — In  the  case  of  the  third  simple  pendulum 
used  (with  the  larger  bob)  what  is  the  percent  error  in  the 
length  due  to  the  assumption  that  the  length  equals  d? 

9.  LAW  OF  FALLING  BODIES. 

The  object  of  this  experiment  is  to  observe  the  time  and 
distance  of  the  free  fall  of  a ball  and,  applying  the  laws  of 
uniformly  accelerated  motion  in  a straight  line,  to  determine 
how  accurate  a value  of  the  acceleration  of  gravity  can  be 
found  from  such  observations.  (D.  26-29,  G.  51,  K.  96-99, 
W.  35.) 

Apparatus. — Falling  body  apparatus,  beam  compass, 
meter  stick,  tray,  steel  ball,  terminals  of  storage  battery  and 
regulating  resistance. 

Theory,  Method  and  Description. — If  a body  falls 
freely  a distance  d in  a time  t,  we  have  the  relation 

d = 1/2  gf 

where  g is  the  acceleration  due  to  gravity.  With  the  appar- 
atus we  measure  corresponding  values  of  d and  t and  since 
g is  constant  we  can  test  the  above  law  by  seeing  whether 
the  ratio  of  corresponding  values  of  d and  t 2 is  constant.  We 
can  also  substitute  corresponding  values  in  the  formula  and 
get  a value  for  g. 

In  the  experiment,  a small  ball  B is  held  by  an  electro- 
magnet A (See  Fig.  17.)  A pendulum  KWP  is  hung  from 


In  General  Physics 


67 


the  knife  edges  K just  under  the  ball,  so  that,  when  the 
latter  is  released,  it  falls  through  a slit  X in  the  platform  P 
at  the  lower  end  of  the  pendulum  when  the  pendulum  is  at 


rest.  The  pendulum  can  be  held  at  one  end  of  its  swing 
by  a second  electromagnet  C.  Both  A and  C are  in  the 
same  electric  circuit,  so  that  when  the  current  is  broken  the 
ball  and  pendulum  are  released  at  the  same  time.  The 
distance  of  fall  can  be  varied  by  raising  or  lowering  the  rod 
which  carries  the  magnet  A,  until  it  is  at  such  a position 


68 


Manual  of  Experiments 


that  the  ball  drops  through  the  slit  X as  the  pendulum  swings 
through  its  position  of  rest.  The  time  of  fall  then  equals 
the  time  required  for  the  pendulum  to  swing  from  one  end 
of  its  path  to  its  position  of  rest,  which  is  one  quarter  of  its 
period.  The  corresponding  values  of  d and  t are  then  to  be 
measured.  The  period  may  be  altered  for  a second  trial 
by  shifting  the  iron  frame  W to  a different  position. 

Directions.— The  battery  circuit  is  connected  through 
a resistance  and  a snap  switch  to  the  binding  posts  on  the 
fall  apparatus.  Adjust  the  current  by  increasing  or  de- 
creasing the  resistance  until  the  electromagnets  are  just 
strong  enough  to  hold  the  ball  and  the  pendulum.  Slip 
the  frame  W to  the  bottom  of  the  pendulum  and  adjust  the 
position  of  the  slit  X so  that  the  ball  drops  through  it  when 
the  pendulum  is  at  rest  and  the  width  of  the  slit  is  not  over 
one  fourth  of  a millimeter  greater  than  the  diameter  of  the 
ball.  Then  by  adjusting  the  height  of  the  upper  magnet, 
find  the  position  for  which  the  ball  will  slip  through  the  slit 
of  the  swinging  pendulum  when  ball  and  pendulum  are  re- 
leased simultaneously  by  snapping  the  switch  quickly . One 
jaw  of  the  slit  is  of  lead,  the  other  of  brass,  so  that  one  can 
tell  by  the  click  on  which  plate  the  ball  strikes,  hence  whether 
it  strikes  in  front  of  or  behind  the  slit.  In  the  first  case  the 
magnet  should  be  raised;  in  the  second  it  should  be  lowered. 
The  tray  should  be  placed  under  the  apparatus  to  catch  the 
ball  as  it  drops.  When  the  height  of  the  ball  has  been  pro- 
perly adjusted,  measure  the  distance  d from  the  center  of  the 
ball  to  the  slit  with  the  beam  compass  and  meter  stick. 
Then  set  the  pendulum  in  vibration  and  determine  the  time 
of  50  complete  swings.  Repeat  this  a second  time  and  cal- 
culate the  mean  quarter  period  of  the  pendulum. 

Change  the  position  of  the  frame  W so  that  the  period 
and  distance  d are  altered,  and  repeat  the  experiment. 


In  General  Physics 


69 


With  these  values  of  d and  t calculate  the  acceleration 
of  gravity  by  the  formula  given. 

Finally  determine  by  about  ten  trials  the  upper  and 
lower  limits  of  the  height  through  which  the  magnet  A can 
be  shifted  and  still  allow  the  ball  to  pass  through  the  slit  in 
the  swinging  pendulum. 

Question. — What  is  the  maximum  possible  percent  er- 
ror in  the  calculated  value  of  g due  to  this  possible  variation 
in  the  height  d determined  by  the  last  measurement? 

10.  ACCELERATION  OF  GRAVITY  WITH  A FALL- 
ING TUNING  FORK. 

To  determine  the. value  of  the  acceleration  of  gravity. 
(D.  26-29;  G.  81;  K.  96-99;  W.  35.) 

Apparatus. — Acceleration  apparatus,  meter  stick. 

Description  and  Theory. — The  acceleration  apparatus 
(Fig.  18a)  consists  essentially  of  a tuning  fork  attached  to  a 
frame  arranged  to  fall  about  one  meter  between  two  vertical 
guides  which  offer  as  little  resistance  as  possible.  The  fork 
is  held  at  the  top  of  the  guides  by  a trigger  by  means  of  which 
it  can  be  released  and  at  the  same  time  set  in  vibration.  On 
one  prong  of  the  tuning  fork  is  a stylus  for  recording  the  vi- 
brations upon  a long  plate  of  smoked  glass  in  the  nature  of 
a curve  such  as  is  shown  in  Fig.  18b. 

If  the  point  a (Fig.  18b)  near  the  beginning  of  the  trac- 
ing, where  the  waves  are  perfectly  formed,  is  selected  and 
spaces  of  exactly  six  wave  lengths  marked  off,  it  is  seen  that 
any  set  of  six  wave  lengths  is  longer  than  the  set  of  six  just 
preceding  it.  The  time  required  for  the  tuning  fork  to  pass 
over  the  distance  is  the  same,  however,  because  the  fork  has 
made  six  complete  vibrations  in  each  case. 


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Manual  of  Experiments 


When  the  fork  had  reached  the  point  a it  had  acquired 
a velocity  which  may  be  represented  by  v0.  If  t represents 
the  time  for  the  fork  to  make  six  complete  vibrations,  it  will 


Fig.  18a  and  b 


pass  over  a distance  vQt  during  the  interval  t because  of  its 
initial  velocity  alone.  But  gravity  also  acts  and  will  in- 
crease the  space  passed  over  in  the  interval  t by  the  amount 
The  total  space  passed  over  in  the  first  interval  a\ob 
is  then, 


In  General  Physics 


71 


ab  = vQt  + 3 4gt2  (1) 

In  two  intervals  of  time  t,  (i.  e.  2 1),  the  tuning  fork  will  pass 
over  a distance  given  by 

ac  = 2v0t  + YigQtY  (2) 

and  for  three  intervals 

ad  = 3v0  t + 3^g(3*)2  ' (3) 

The  differences  ab , ac  — ab  and  ad—ac  are  the  distances 
passed  over  in  successive  equal  intervals  of  time.  The 
distances  are  given  by  the  following  equations: 

ab  = vQt  + Yzg?  (1) 

ac  — ab  = bc  = v0t  + | gt2  (4) 

ad  — ac  = cd  = vQt  + | gt 2 (5) 

From  these  equations  it  is  seen  that  the  increase  in  the  space 
passed  over  in  successive  intervals  is 

bc  — ab  = cd  — bc  = gt 2 (6) 

This  states  that  the  difference  between  the  distances  repre- 
sented by  any  two  successive  groups  of  six  wave  lengths  is 
given  by  the  value  of  gravity  times  the  square  of  the  time 
consumed  by  the  tuning  fork  in  making  six  complete  vi- 
brations. The  value  of  g,  the  acceleration  of  gravity,  can  be 
determined  if  the  time  t required  for  the  fork  to  make  six 
complete  vibrations  and  the  difference  between  the  distances  , 
passed  over  in  two  successive  six  vibration^  intervals  are 
known.  The  time  t is  obtained  by  dividing  the  number  of 
vibrations  made  in  the  time  t by  the  number  made  in  one 
second  which  is  recorded  on  the  fork,  or 

t =1  (7) 

where  n is  the  pitch  of  the  fork.  The  difference  between 
successive  distances  can  be  obtained  by  direct  measurement. 

Directions. — In  order  that  the  friction  may  be  as  little 
as  possible  the  uprights  should  be  standing  vertically.  Put 


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Manual  of  Experiments 


the  smoked  glass  plate  in  position  moving  it  towards  one 
side  so  that  the  first  tracing  made  will  be  near  one  edge. 
Fasten  the  tuning  fork  on  the  eccentric  at  the  top  of  the 
frame  and,  having  the  stylus  pressing  closely  against  the 
glass  plate,  turn  the  lever  connected  to  the  eccentric  so  as 
to  release  the  fork.  The  fork  will  fall  freely  and,  in  so  doing, 
make  a tracing  upon  the  smoked  glass.  In  a similar  manner 
make  at  least  six  tracings  on  the  plate,  shifting  the  position 
of  the  plate  each  time  so  as  to  have  each  tracing  separate 
from  the  others.  Remove  the  plate  from  the  frame  and, 
beginning  with  the  first  good  wave,  mark  off  spaces  of  six 
wave  lengths  on  each  tracing.  Measure  with  a millimeter 
scale  all  of  the  six  vibration  spaces  on  each  tracing  that  it 
is  possible  to  obtain,  and  find  the  difference  between  suc- 
cessive distances  represented  by  six  wave  lengths.  Find  the 
mean  value  of  the  differences  for  each  of  the  tracings. 

Note  the  pitch  of  the  fork  as  recorded  upon  it  and,  sub- 
stituting the  pitch  n in  equation  (7),  find  the  value  of  t. 
Substitute  the  mean  value  of  the  differences  for  one  of  the 
waves  and  the  value  of  t in  equation  (6)  and  solve  for  g.  In 
like  manner  determine  the  value  of  g as  given  by  the  data  of 
each  of  the  other  curves.  Calculate  the  mean  of  the  values 
of  g thus  found. 

Questions. — What  effect  will  friction  of  the  falling 
frame  have  on  the  value  of  g?  Explain  your  answer.  What 
percent  error  in  g will  be  caused  by  an  error  of  one  percent 

in  n,  the  frequency  of  the  fork? 

■L- 

11.  FRICTION. 

To  study  the  laws  of  sliding  friction  and  to  determine 
the  coefficients  of  static  and  of  kinetic  friction.  (D.  126- 
130,  G.  49,  K.  80-84,  W.  96-99). 


-j 


In  General  Physics 


73 


Apparatus. — Friction  table,  block  of  wood,  strip  of  al- 
uminum or  glass,  coarse  balance  and  weights,  small  can  and 
thread,  metal  shot  and  spirit  level. 

Method,  Theory  and  Description. — By  friction  we 
mean  the  resistance  to  motion  between  two  bodies  in  con- 
tact, caused  by  their  surfaces.  This  resistance  is  independ- 
ent of  the  area  of  contact  between  the  two  bodies  but  is  di- 
rectly proportional  to  the  total  pressure  between  them.  In 
general,  friction  between  two  surfaces  of  the  same  substance 
is  greater  than  friction  between  two  surfaces  of  different  sub- 
stances. When  the  two  bodies  are  at  rest,  the  force  neces- 
sary to  make  one  slide  horizontally  upon  the  other  must 
exceed  a certain  definite  value,  which  is  called  the  1 1 maximum 
static  friction  The  ratio  of  the  maximum  static  friction  F 
between  two  surfaces , to  the  normal  pressure  N between  them 
is  called  the  “ coefficient  of  static  friction 

After  the  one  body  has  started  to  slide  over  the  other  a 
certain  force  is  necessary  to  keep  it  in  uniform  horizontal 
motion.  The  ratio  of  this  friction  force  to  the  total  pressure  is 
called  the  “coefficient  of  kinetic  friction ” p. 

When  one  of  the  surfaces  is  a plane  inclined  to  the  hor- 
izontal and  the  angle  of  tilt  is  gradually  increased,  there  is 
a critical  angle  tli”  at  which  the  second  surface  will  slide 
uniformly  down  the  plane  when  given  a start.  It  may  be 
shown  that  the  tangent  of  this  critical  angle  is  numerically 
equal  to  the  coefficient  of  kinetic  friction , or 

tan  i = p 

The  friction  table  Fig.  19  consists  of  a heavy  base  board 
mounted  on  leveling  screws,  upon  which  a lighter  board  with 
a very  smooth  top  is  hinged  at  one  end.  By  means  of  the 
hinges  and  a side  clamp,  the  upper  board  may  be  fixed  at 
any  desired  angle  to  the  base  board.  A small  block  of  wood 


74 


Manual  of  Experiments 


with  four  smooth  sides  is  made  to  slide  on  the  upper  board. 
(Two  of  these  sides  have  twice  the  area  of  the  other  two). 
It  is  pulled  along  by  a weight,  hung  from  a thread  which 
passes  over  a universal  pulley  clamped  at  the  end  of  the  table. 

The  friction  between  the  two  surfaces  under  the  different 
conditions  mentioned  above  and  the  corresponding  pressures 
between  them  are  observed,  and  the  coefficient  of  friction 
calculated  in  each  case. 


faces  with  the  hands.  The  oil  of  the  skin  will  lubricate  the 
spots  touched  and  spoil  the  uniformity  of  the  surfaces. 

Place  the  top  board  flat  upon  the  base  and  get  its  upper 
surface  level.  To  do  this  place  the  level  in  the  center  with 
its  length  across  the  board,  and  adjust  the  two  leveling 
screws  at  one  end  of  the  board  until  the  bubble  of  air  in  the 
level  is  at  the  center  mark.  Then  place  the  spirit  level 
lengthwise  of  the  board  and  adjust  the  third  leveling  screw 
until  the  bubble  stays  at  the  center  mark.  (In  every  case 
below,  adjust  the  pulley  until  the  thread  is  parallel  to  the 
upper  board.) 

Static  and  kinetic  friction. — Fasten  one  end  of  the  thread 
to  the  hook  on  the  block,  pass  it  over  the  pulley,  and  fasten 


In  General  Physics 


75 


the  other  end  to  the  bucket.  Draw  the  block  to  the  far  end 
of  the  friction  table  with  its  flat  side  on  the  board,  and  place 
a 100  gram  mass  on  top  of  it.  Then  carefully  place  shot  in 
the  bucket  until  the  block  just  begins  to  slide.  The  observer 
should  hold  one  hand  under  the  bucket  while  dropping  in 
the  shot,  to  prevent  spilling  the  latter  if  the  bucket  begins 
to  drop  suddenly.  Owing  to  the  fact  that  the  friction  table 
is  not  uniformly  smooth  all  over  its  surface,  or  that  the  re- 
lative directions  of  the  grain  of  block  and  board  are  not  ex- 
actly the  same  in  successive  trials,  or  to  other  causes,  the 
amount  of  shot  needed  may  vary  in  successive  trials.  Do 
not  spend  too  much  time  in  trying  to  get  such  trials  to  agree. 
Repeat  the  trial  three  or  four  times  and  take  the  mean  weight 
of  bucket  and  added  shot  for  the  force  necessary  to  over- 
come the  friction. 

Under  the  same  conditions  vary  the  amount  of  shot  in 
the  bucket  and  find  what  weight  of  bucket  and  shot  (the 
mean  of  several  trials)  is  necessary  to  keep  the  block  moving 
uniformly  after  a slight  push  has  been  given  to  start  it. 
Find  the  weight  of  block  and  added  100  grams  mass,  and 
from  the  data,  calculate  the  coefficients  of  static  and  of 
kinetic  friction  and  compare  them. 

Friction  corresponding  to  different  areas  of  contact. — Turn 
the  block  on  edge  and  adjust  the  pulley  until  the  thread  is 
parallel  to  the  upper  board.  Place  the  100  gram  mass  on 
the  block  and  find  as  before  what  weight  is  necessary  to 
keep  it  in  uniform  motion  after  a start  has  been  given  it. 
Measure  the  areas  of  the  two  surfaces  of  the  block  that  were 
used  as  friction  surfaces.  Compare  the  kinetic  friction  in 
this  trial  with  that  in  the  previous  trial  and  note  what  effect 
the  area  of  the  surface  of  contact  has  upon  the  friction. 

V ariation  of  friction  force  F with  total  normal  pressure 
N. — Find  the  friction  force  when  the  block  is  laid  flat  with 


76 


Manual  of  Experiments 


100  grams  mass  on  it.  Repeat  with  300  grams  mass  on  it. 
Show  from  your  data  that  F is  directly  proportional  to  N. 
(Remember  that  N includes  the  weight  of  the  block  in  each 
case).  Compare  the  coefficients  of  kinetic  friction  in  the 
two  cases. 

Variation  of  friction  force  with  nature  of  surface  of  con- 
tact.— Place  the  strip  of  aluminum  (or  glass)  on  top  of  the 
friction  board,  holding  the  former  by  the  edges  while  hand- 
ling it.  Determine  the  mass  necessary  to  slide  the  block 
with  100  grams  on  it  uniformly  along  the  surface  of  the  metal. 
Calculate  the  coefficient  of  friction  and  compare  it  with  that 
between  wood  and  wood. 

To  determine  the  critical  angle. — Remove  the  aluminum 
plate,  and  take  the  thread  from  the  block.  Tilt  the  top 
board  to  such  an  angle  that  when  started  by  a slight  push, 
the  block  will  slide  uniformly  down  the  board.  Clamp  the 
board  in  this  position  and  measure  the  perpendicular  “a” 
dropped  from  the  lower  face  of  the  upper  end  of  the  board 
upon  the  base,  and  the  distance  iCb”  from  the  hinged  end 
of  the  board  to  this  perpendicular.  See  Fig.  20.  The  ratio 


a/b  is  equal  to  the  tangent  of  the  critical  angle.  Compare 
this  tangent  with  the  coefficient  of  friction  4>  found  for  the 
flat  block  and  100  grams  mass  sliding  on  the  horizontal 
wooden  surface  with  uniform  motion.  * 


In  General  Physics 


77 


Question. — Prove  with  the  aid  of  a diagram  the  fact 
that  tangent  i=  </>,  and  show  that  ‘ Y ” is  independent  of  the 
mass  of  the  block. 


■ 


12  STRETCHING  WIRES  AND  YOUNG  ’S  MODULUS 


To  study  laws  of  stretching  wires  and  to  determine 
Young ’s  modulus  of  elasticity.  (D.  164-172,  G.  20,  87. 
K.  235-239,  244,  W.  122,  17-2.) 

Apparatus. — Stretching  wire  apparatus  with  optical 
lever,  telescope  and  scale,  four  2Kg.  and  two  lKg.  weights, 
micrometer  calipers,  beam  compass,  meter  stick,  weight 
holder,  and  three  wires,  (two  of  the  same  diameter  but  dif- 
ferent lengths,  and  a third  of  the  same  length  as  one  of  these 
but  having  a different  diameter.) 

Theory,  Description  and  Method: — The  elasticity 
of  a body  is  its  tendency  to  recover  from  distortion  when  the 
forces  which  produce  the  strain  are  removed.  By  strain  we 
mean  the  change  of  shape  or  volume  produced  by  the 
forces.  By  stress  we  mean  the  internal  forces  between 
contiguous  parts  of  the  body  when  it  is  in  a state  of  strain. 
Hook’s  law  states  that  within  the  elastic  limits , stress  is  pro- 
portional to  strain.  The  measure  or  modulus  of  elasticity  is 
the  ratio  of  the  magnitude  of  the  stress  to  that  of  the  accompany- 
ing strain. 

In  the  case  of  simple  longitudinal  strain,  as  for  instance, 
a stretching  wire,  this  modulus  is  known  as  Young’s  modul- 
us. In  this  case  the  stress  is  equal  to  the  external  force  per 
unit  area  of  cross  section  of  wire,  or  =F /a  where  F is  the 
force  and  a the  area.  The  strain  or  stretch  is  the  extension 
per  unit  length  of  wire,  or  =l/L  where  / is  the  total  extension 
and  L the  length.  Then  for  a circular  wire  of  radius  r,  the 
area  is  7r r2  and  Young  modulus  is 


78 


Manual  of  Experiments 


stress  F / FL 

Y = — — — = — = — 

strain  a L t r2l 

The  apparatus  consists  of  a supporting  frame  of  two 
heavy  steel  rods  about  150  cm.  long,  mounted  vertically  on 
a tripod  with  leveling  screws,  and  fitted  with  a clamp  at  the 


top  for  bolding  the  wire  to  be  tested.  See  Fig.  21.  A small 
bushing  is  clamped  on  the  wire  near  its  lower  end,  and  an 
adjustable  platform,  sliding  on  the  supporting  rods,  may  be 
clamped  at  the  same  height  as  this  bushing.  The  wire  is 


In  General  Physics 


79 


stretched  by  means  of  weights  which  are  placed  on  the  sup- 
port hung  from  its  l6wer  end,  and  the  amount  of  the  stretch 
is  measured  by  means  of  an  optical  lever  and  telescope  with 
vertical  scale  attached.  The  optical  lever  consists  of  a T 
shaped  base  with  a leg  at  each  extremity  and  a small  plane 
mirror  mounted  on  a horizontal  axis  above  the  short  arm  of 
the  T and  facing  away  from  the  third  leg.  This  leg  rests 
on  the  bushing  clamped  on  the  wire.  The  other  legs  rest 
on  the  platform.  In  its  normal  position  the  plane  of  the 


mirror  is  vertical  and  the  telescope  is  mounted  directly  in 
front  of  it  at  a distance  of  about  80  cm.,  so  that  on  looking 
through  the  telescope  when  it  is  properly  adjusted,  we  see 
the  center  of  the  scale  reflected  in  the  mirror.  If  now  the 
wire  is  stretched,  the  leg  V is  lowered,  (see  Fig.  22)  tilting 
the  lever  through  an  angle  c/>.  The  mirror  and  its  normal 
N M will  also  be  tilted  through  this  angle  </>.  (In  the  figure 
M is  misplaced  from  the  intersection  of  lines  from  S,  N and 
T.)  But  by  the  law  of  reflection  of  light  rays,  the  angle 
made  by  the  incident  ray  (from  scale  S to  mirror  M)  with 
the  normal  to  the  mirror  (MN)  is  equal  to  the  angle  be- 
tween the  normal  and  the  reflected  ray  (from  the  mirror  M 
to  the  telescope  T),  and  by  the  geometry  of  the  figure  we 
see  that  these  angles  equal  </>  also.  The  point  S on  the  scale 
will  therefore  be  seen  in  the  telescope.  For  the  small 
angles  of  tilt  to  be  used,  the  point  of  intersection  N between 


80 


Manual  of  Experiments 


normal  and  scale  may  be  considered  as  halfway  between  S 
and  T,  and  NM  as  equal  in  length  to  TM.  Then  in  the 
similar  triangles  VCO  and  NMT  we  have 

VO  TN  VC  x TN 

= — — , or  VO  = — 

VC  TM  TM 

where  VO  is  the  stretch  of  the  wire,  VC  is  the  perpendicular 
distance  from  the  leg  V to  the  line  joining  the  other  two 
legs,  TN  is  half  the  distance  between  the  points  on  the  scale 
observed  in  the  telescope  before  and  after  tilting  the  mirror, 
and  TM  is  the  distance  from  scale  to  mirror. 

Directions. — Caution. — In  order  to  prevent  bending 
and  kinking  of  the  wires,  keep  them  stretched  by  a small 
mass.  The  wire  when  not  in  use  should  be  hung  on  the 
supports  provided. 

Preliminary. — Place  the  optical  lever  on  a sheet  of  paper 
and  press  lightly  on  it,  leaving  an  imprint  of  the  three  legs. 
Draw  a straight  line  between  the  two  nearer  marks  and  drop 
a perpendicular  from  the  third  mark  upon  this  line,  and 
measure  its  length.  Measure  also  the  diameters  of  the 
three  wires  with  the  micrometer  calipers,  and  the  effec- 
tive stretching  length  between  bushing  and  lug  with  the 
beam  compass  and  meter  stick.  After  the  stretching 
weight  is  removed,  the  wire  to  be  used  is  slipped  through 
the  hole  in  the  platform  and  fastened  in  the  upper 
clamp  by  means  of  the  lug.  To  free  the  wire  of  all  bends 
an  initial  stretching  weight  of  2Kg.  is  hung  on  it , and  all  no 
load  readings  are  to  be  taken  with  this  weight  attached.  The 
platform  is  placed  in  line  with  the  bushing  on  the  wire,  and 
the  tripod  is  leveled  so  that  the  bushing  slips  freely  through 
the  opening  when  the  wire  is  stretched.  The  optical  lever 
is  then  placed  on  bushing  and  platform.  The  telescope  and 
platform  must  be  adjusted  until  the  middle  of  the  scale  is 


In  General  Physics 


81 


seen  in  the  former.  The  telescope  has  three  tubes  which 
slide  within  each  other,  one  containing  the  object  lens,  the 
second  a pair  of  cross  hairs,  and  the  third  the  eyepiece.  The 
latter  must  be  drawn  out  until  the  cross  hairs  are  seen  sharp- 
ly focussed;  then  the  second  tube  is  drawn  out  of  the  first 
until  there  is  no  parallax  between  the  cross  hairs  and  the 
scale  divisions  seen  in  the  telescope;  i.  e.  until  there  is  no 
relative  motion  between  the  two  when  the  eye  is  moved 
from  side  to  side  in  front  of  the  eyepiece.  This  adjustment 
should  not  be  disturbed  during  the  remainder  of  the  experi- 
ment. The  distance  from  mirror  to  center  of  scale  should 
then  be  measured. 

Relation  between  stretch  and  length  of.  wire. — Suspend  the 
shorter  wire  with  a weight  of  5Kg.  and  note  the  reading  at 
the  cross  hairs  in  the  telescope.  Then  carefully  remove  three 
kilograms,  the  stretching  weight,  and  note  the  no  load  read- 
ing in  the  telescope.  Replace  the  wire  by  a longer  one  of 
the  same  diameter,  and  note  the  no  load  and  load  readings 
for  the  same  stretching  force.  Then  calculate  the  stretch 
by  the  formula  given,  and  the  ratio  of  stretch  to  original 
length  for  both  trials,  and  compare  them. 

Relation  between  stretch  and  cross  sectional  area  of  wire. — • 
Replace  the  last  wire  used  by  the  third  one,  having  the  same 
length  but  a different  diameter,  and  take  the  load  and  no 
load  readings  with  the  same  stretching  force  as  before.  Cal- 
culate the  stretch  and  the  cross  sectional  areas  of  the  last 
two  wires  used.  Calculate  also  the  product  of  stretch  and 
area  for  each  wire,  and  compare  these  products. 

Relation  between  stretch  and  stretching  force , Hook’s  law — 
Using  the  same  wire  as  in  the  last  case,  change  the  stretching 
force  to  5Kg.  and  note  the  load  and  no  load  readings. 
Calculate  the  stretch  as  before.  Calculate  also  the  ratio  of 


¥i 


< 


82 


Manual  of  Experiments 


the  stretch  to  the  stretching  mass  for  each  of  the  two  trials 
with  this  last  wire,  and  compare  these  ratios. 

Young7 s modulus . — From  the  data  taken  in  the  last  case 
calculate  the  value  of  Young’s  modulus  for  the  wire  used 
(brass  or  steel)  Assume  g = 980  cm/sec2. 

Question. — The  data  taken  show  that  the  stretch 
/ = CmL/ A,  where  m is  the  mass  producing  the  stretch,  and 
C is  a constant.  What  is  the  value  of  this  constant? 


13.  LAWS  OF  BENDING  OF  RODS. 
.Modulus  of  Elasticity. 

To  study  the  relation  between  the  bend  of  a rod  and  its 
length,  breadth, depth, and  material,  and  the  bending  force; 
also  to  calculate  Young’s  modulus  for  brass  and  steel.  (D. 
164-173;  G.  20,  88;  K.  235-239,  244;  W.  122,  173.) 


Apparatus. — Channel  iron  bench  about  a meter  long, 
micrometer  screw,  micrometer  calipers,  two  blocks  with  knife 
edges,  two  weights,  weight  holder  and  loop  of  thread,  meter 
stick,  and  three  rods,  two  of  them  having  the  same  square 
cross  sections  but  one  of  brass,  the  other  of  steel,  and  the 
third  of  steel  with  a rectangular  cross  section. 

Theory,  Description  and  Method. — (Read  that  part 
of  the  theory  in  experiment  12  which  deals  with  elasticity.) 
It  has  been  found  that  when  a rod  is  bent  out  of  its  original 
position,  the  amount  of  the  bend  B is  directly  proportional 
to  the  cube  of  its  length  L and  to  the  force  F producing  the 
bend,  and  inversely  proportional  to  the  cube  of  its  depth  d 
and  to  its  breadth  b.  That  is 

FU 

B = C 

bd3 


In  General  Physics 


83 


where  C is  a constant  depending  upon  the  mode  of  support 
and  the  material  of  the  rod.  If  the  rod  be  supported  at  both 
ends  and  the  force  be  applied  midway  between  them,  this 
constant  C = i/(4Y)  where  Y is  the  modulus  of  elasticity  of 
the  material. 

In  this  case  it  is  readily  seen  that  the  lower  layers  of  the 
rod  are  stretched  and  the  upper  layers  compressed.  The 
modulus  of  elasticity  involved  is  therefore  Young’s  modulus. 
The  relation  between  bend  and  modulus  is 

FU  FU 

B = , or  Y = 

4 Ybd3  4:Bbd3 

To  study  the  laws  of  bending  a comparison  is  made  be- 
tween the  values  of  the  bend  obtained  for  different  values 
of  one  of  the  factors  upon  which  it  depends,  all  the  other 
factors  remaining  constant.  The  relation  between  the  bend 
and  this  factor  can  then  be  determined.  If  for  instance  we 
determine  the  amount  of  bend  corresponding  to  two  or  three 
different  values  of  the  length  when  all  other  factors  are  kept 
constant,  and  find  on  examination  that  the  ratio  between 
corresponding  values  of  bend  and  length  cubed  is  constant, 
it  shows  that  the  bend  is  directly  proportional  to  the  cube 
of  the  length,  etc.  For  these  observations  we  use  an  iron 
bench  at  the  middle  of  which  is  a vertical  support  for  a mi- 
crometer screw.  See  Fig.  23.  Two  knife  edges  placed  across 


Fig.  23 


$ ' 


$ 


84 


Manual  of  Expeeiments 


the  bench,  equidistant  from  the  screw,  support  the  rod  to  be 
used.  The  micrometer  is  adjusted  until  its  tip  just  touches 
the  rod,  first  when  the  latter  is  unloaded  except  for  the  wire 
frame,  then  when  it  is  loaded  with  a known  mass  placed  on 
the  frame.  A reading  of  the  micrometer  is  taken  in  each  of 
these  positions,  and  the  difference  between  the  two  readings 
gives  the  bend  of  the  rod. 

The  micrometer  screw  has  a pitch  of  a half  millimeter. 
Its  straight  scale  is  divided  into  millimeters,  its  circular  scale 
into  100  equal  divisions  so  that  each  division  represents 
1/100  of  1/2  millimeter  of' 1/2000  cm.  To  avoid  errors  in 
reading  the  micrometer  the  following  method  may  be  used: 
Adjust  the  micrometer  until  the  screw  tip  is  about  1/8  inch 
above  the  rod,  the  zero  of  the  disc  coinciding  with  one  edge 
of  the  vertical  scale.  Carefully  turn  the  screw  until  the  tip 
just  touches  the  rod,  counting  the  number  of  complete  re- 
volutions and  observing  the  fractional  part  of  a revolution 
(the  number  of  divisions  beyond  zero  through  which  the  disc 
is  turned),  estimating  to  1/10  division.  Thus  for  two  com- 
plete turns  and  10  and  1/10  divisions  beyond  zero  the  read- 
ing would  be  210.1  divisions.  Call  this  reading  R0.  Return 
the  screw  to  its  original  position,  suspend  a mass  from  the 
rod,  and  proceed  as  before  taking  the  new  reading  Ri  when 
the  screw  just  touches  the  rod.  The  bend  in  cm.  will  be 
(Ri-Ro)  /2000. 

It  is  somewhat  difficult  to  see  just  when  the  point  of  the 
screw  comes  in  contact  with  the  rod,  therefore  an  electric 
buzzer  B is  used  as  a detector.  See  Fig.  23.  A dry  cell  is 
connected  by  means  of  wires  to  the  screw  and  to  the  buzzer 
and  from  the  latter  to  one  end  of  the  rod.  Hence  when  the 
screw  comes  in  contact  with  the  rod,  the  circuit  will^be 
closed  and  the  current  of  electricity  from  the  cell  flowing 
through  the  buzzer  will  cause  it  to  emit  a humming  noise. 


In  General  Physics 


85 


The  tip  of  the  screw  and  the  part  of  the  rod  which  it  touches 
must  be  kept  well  polished  to  offer  as  little  resistance  to  the 
flow  of  current  as  possible,  otherwise  the  detector  will  not 
buzz  until  there  is  considerable  pressure  between  screw  and 
rod. 

Directions. — Relation  between  bend  and  length  of  rod , 
and  between  bend  and  force  producing  the  bend. — Polish  the 
middle  of  the  rods  where  the  screw  tip  will  touch  them. 
Place  the  knife  edges  80  cm.  apart  with  the  micrometer  half 
way  between  them  and  lay  the  small  steel  rod  flat  on  them. 
Lay  the  loop  of  thread  over  the  middle  of  the  rod,  bring 
both  ends  down  through  the  hole  in  the  bench,  and  hook 
them  together  with  the  wire  frame  which  is  to  support  the 
li weights. M Adjust  the  micrometer  and  have  the  threads 
about  5mm.  apart  so  that  the  screw  can  touch  the  rod  be- 
tween them.  Turn  the  screw  until  its  tip  just  meets  its 
image  reflected  from  the  rod.  The  humming  of  the  buzzer 
will  indicate  when  this  occurs,  if  it  cannot  be  seen  with  the 
eye.  Read  the  micrometer  to  1/10  division  and  repeat 
twice  more,  averaging  the  three  readings  to  get  R0.  Then 
hang  the  smaller  mass  on  the  wire  frame,  take  an  average 
of  three  readings  for  Ri,  and  calculate  the  bend  for  the 
weight  of  this  mass,  wj.  In  the  same  way  find  the  bend 
corresponding  to  the  weight  of  a larger  mass,  m2.  Then 
move  the  knife  edges  closer  until  they  are  60  cm.  apart, 
the  micrometer  still  being  midway  between  them,  and  find 
the  bend  under  the  weight  of  the  larger  mass.  From  these 
data  show  that  the  bend  varies  directly  as  the  force  applied 
and  as  the  cube  of  the  length  between  supports  by  calculat- 
ing the  ratios  of  corresponding  values  of  bend  and  mass  pro- 
ducing the  bend  for  each  case  tried,  and  comparing  them; 
and  of  bend  and  length  cubed  for  each  case  and  comparing 
them. 


86 


Manual  of  Experiments 


Relation  between  bend  and  breadth  of  rod , and  between  bend 
and  depth  of  rod. — Measure  with  the  micrometer  calipers 
the  width  and  thickness  of  all  three  rods.  Take  several 
measurements  at  different  parts  of  the  rod  and  average 
them.  Using  a length  of  80  cm.  and  the  larger  mass  de- 
termine the  bend  of  the  larger  steel  rod,  first  with  its  flat 
side  (the  breadth)  on  the  knife  edges,  then  with  its  narrow 
side,  (which  was  the  depth  in  the  previous  trial)  on  knife 
edges.  If  time  permits,  redetermine  the  bend  of  the  small 
steel  rod  for  the  same  length  and  mass;  if  not  use  the  data 
already  obtained  for  this  case.  Show  with  your  data  that 
the  bend  varies  inversely  as  the  breadth  and  inversely  as  the 
cube  of  the  depth  of  the  rod,  by  calculating  the  products  of 
corresponding  values  of  bend  and  breadth  for  each  case 
tried,  and  comparing  them;  and  of  bend  and  depth  cubed 
for  each  case  and  comparing  them. 

Variation  of  bend  with  material  of  rod.  Y oung ’s  modulus. — - 
Determine  the  bend  of  the  brass  rod  for  a length  of  80  cm. 
and  the  weight  of  the  small  mass.  Compare  this  with  the 
bend  of  the  small  steel  rod  under  the  same  conditions  and 
assuming  that  they  have  the  same  breadth  and  depth,  find 
the  ratio  of  the  bends,  that  of  steel  being  taken  as  1. 

From  the  data  for  the  first,  and  last  trials  calculate 
Young’s  modulus  for  steel  and  for  brass. 

Assume  g = 980  cm.  /sec2. 

Question. — Using  the  formula  given  for  the  bend  of  a 
rod,  what  is  the  simplest  expression  for  the  ratio  of  the  bend 
of  one  rod  to  that  of  another?  (Distinguish  between  the 
various  factors  for  the  two  rods  by  means  of  subscripts  1 
and  2.) 


In  General  Physics 


87 


14.  LAWS  OF  TWISTING  OF  RODS 
Modulus  of  Rigidity. 

To  study  the  relation  between  the  amount  of  twist  of  a 
rod  and  its  length,  its  diameter,  the  force  producing  the 
twist,  and  the  nature  of  the  rod;  also  to  find  the  modulus  of 
rigidity  of  steel  and  of  brass.  (D.  164-170;  G.  20,  89;  K. 
235-239,  243;  W.  122,  171,  174.) 

Apparatus. — Iron  bench,  micrometer  calipers,  two 
weights,  meter  stick,  and  four  rods,  three  steel  ones  of  dif- 
ferent diameters,  and  one  brass  one. 

Theory,  Method  and  Description. — (See  that  part 
of  theory  in  experiment  12  which  deals  with  elasticity.) 
It  has  been  found  that  the  amount  of  twist  in  a metal  rod 
is  directly  proportional  to  its  length  and  to  the  force  pro- 
ducing the  twist  and  inversely  proportional  to  the  fourth 
power  of  its  radius.  The  rigidity  of  a rod  under  the  action 
of  a certain  force  is  inversely  proportional  to  the  twist  pro- 
duced by  the  force.  That  is  if  R and  T are  respectively  the 
rigidity  and  the  angle  through  which  the  rod  is  twisted,  the 
relative  rigidity  of  two  rods  will  be  Ri/R2  = T2/ 1\. 

When  a rod  is  twisted  the  distortion  is  a change  of  shape 
only;  that  is,  it  has  undergone  a shear.  The  ratio  of  the 
shearing  stress  per  unit  area  to  the  shearing  strain  per  unit 
length  is  the  “modulus  of  rigidity  ” (sometimes  called  co- 
efficient of  torsional  rigidity  or  simple  rigidity.)  It  may  be 
shown  that  this  modulus  is 

. 2 LM 

n = 

7 r0r4 

where  L is  the  length  of  the  rod,  r its  radius,  M the  moment 
of  the  torsional  couple,  and  6 the  angle  (in  radians)  through 


88 


Manual  of  Experiments 


which  the  rod  is  twisted.  If  the  rod  is  twisted  by  a mass  m 
hanging  with  a moment  arm  R , we  have  M = mgR  and 

2 LmgR 

n = 

7r  dr 4 

To  study  the  relation  between  the  twist  of  a rod  and  any 
of  the  factors  upon  which  the  twist  depends,  the  twist  for 
different  values  of  this  factor  are  obtained,  all  other  factors 
remaining  constant.  From  a comparison  between  a value 
of  the  factor  and  corresponding  value  of  the  twist,  the  law 
of  variation  between  them  can  be  determined.  Keeping 
all  factors  constant  but  using  two  rods  of  different  materials 
and  observing  the  twist  for  a given  force,  their  relative  rigi- 
dity can  be  determined  by  calculating  the  inverse  ratio  of 
their  twists. 


1 

■L\ 

L.  . „ - 

Si 

p ® j 

— oot  , 

— 

1 

— — — 

Fig.  24 


The  torsion  bench  is  about  a meter  long.  See  Fig.  24. 
An  iron  clamp  may  be  screwed  fast  at  various  positions  on 
the  bench  and  serves  to  hold  in  a fixed  position  the  rod  to 
be  twisted.  The  end  of  the  rod  is  held  by  a clutch  mounted 
on  ball  bearings  at  one  end  of  the  bench.  Fastened  to  the 
clutch  is  a grooved  wooden  wheel  from  the  circumference  of 
which  is  to  be  suspended  the  mass  which  produces  the  twist. 


In  General  Physics 


89 


A circular  scale  is  attached  to  the  wheel  and  a fixed  pointer 
indicates  on  the  scale  the  angle  through  which  the  rod  has 
been  twisted. 

Directions. — Relation  of  angle  of  twist  to  length  and  to 
force  producing  twist. — Clamp  the  smallest  steel  rod  in  posi- 
tion for  twisting,  having  distance  between  the  clamps  about 
one  meter.  (Some  of  the  small  steel  rods  are  copper  plated 
and  so  do  not  look  like  steel.)  Take  the  initial  or  no  load 
reading  T0  to  0.1  division  on  the  scale  when  only  supporting 
cord  and  hook  hang  from  the  wheel.  Then  suspend  the 
smaller  mass  from  the  wheel  and  note  the  load  reading  T±. 
As  a check  against  possible  slipping  the  no  load  reading 
should  always  be  repeated  after  a load  reading  has  been 
taken.  The  difference  between  these  readings  gives  the  an- 
gle of  twist,  T°.  Replace  the  small  weight  by  the  larger  one 
and  find  the  twist  produced.  Then  change  the  position  of 
the  movable  clamp  so  that  the  rod  is  about  half  its  original 
length  and  find  the  twist  produced  by  the  larger  mass.  Note 
the  weight  of  the  masses  and  measure  the  length  of  the  rod 
between  the  two  lugs  in  each  case  and  from  the  greater 
length  subtract  the  length  of  the  middle  lug  (since  the  part 
of  the  rod  within  this  lug  is  not  appreciably  twisted.)  From 
the  data  taken,  show  how  the  twist  varies  with  the  length 
and  with  the  weight  which  produces  the  twist,  by  comparing 
the  ratios  of  corresponding  values  of  twist  and  length  in 
each  case,  and  the  ratios  of  corresponding  values  of  twist 
and  twisting  mass  in  each  case. 

Relation  between  twist  and  radius  of  rod. — Determine  the 
twist  for  the  next  larger  rod,  using  a length  of  about  one 
meter  and  the  larger  mass.  Do  the  same  for  the  largest 
steel  rod  under  the  same  conditions.  Measure  the  mean 
diameters  of  the  two  rods  with  the  micrometer  calipers,  and 
from  your  data  show  the  relation  between  the  twist  and  the 


90 


Manual  of  Experiments 


radius  of  the  rod,  by  comparing  the  products  of  correspond- 
ing values  of  twist  and  the  fourth  power  of  the  radius  in  each 
case. 

Relative  rigidity  of  steel  and  brass. — Determine  the  twist 
of  about  one  meter  of  brass  rod  for  the  larger  mass.  If 
time  permits  redetermine  the  twist  of  the  largest  steel  rod 
under  the  same  conditions;  if  not,  use  the  previous  deter- 
mination, and  calling  the  rigidity  of  the  steel  rod  unity,  cal- 
culate the  relative  rigidity  of  the  brass  rod. 

Modulus  of  rigidity. — Measure  the  circumference  of  the 
grooved  wheel  with  a thread  and  meter  stick,  and  calculate 
its  radius  R.  Reduce  the  angle  of  twist  in  the  last  two  ob- 
servations from  degrees  to  radians.  Then  from  data  taken 
calculate  the  modulus  of  rigidity  for  steel  and  brass.  As- 
sume g = 980  cm.  /sec2. 

Question. — If  one  rod  has  a times  the  length  and  1/b 
times  the  diameter  of  a second  rod,  and  c times  its  modulus 
of  rigidity,  what  is  the  ratio  of  the  twist  of  the  first  to  that 
of  the  second  when  the  masses  acting  with  the  same  moment 
arm  for  both,  are  in  the  ratio  d to  f respectively? 


15.  MOMENT  OF  INERTIA. 

To  find  the  moment  of  inertia  of  a rectangular  bar  of 
steel.  (D.  81-92;  G.  89  A;  K.  135-140;  W.  85.)  I 

Apparatus. — Upright  support  with  two  pulleys,  ro- 
tating disk,  rectangular  bar,  two  masses  (45  and  100  g.) 
coarse  balance  and  weights,  beam  compass,  two  meter 
sticks,  and  string. 

Theory,  Method,  and  Description. — The  moment  of 
inertia  of  a body  about  an  axis  is  the  sum  of  the  products  of 
the  mass  of  each  particle  of  the  body  into  the  square  of  its 


In  General  Physics 


91 


distance  from  the  axis  of  rotation.  If  / be  the  moment  of 
inertia,  mh  ra2,  etc.,  the  masses  of  different  particles  and 
ri,  r2,  etc.,  their  respective  distances  from  the  axis  of  rotation, 
then 

/ = rmri  + m^rf  + etc. 

In  the  equations  of  nation  for  a body  in  rotation  its  moment 
of  inertia  is  the  equivalent  of  its  mass  in  the  equations  for 
translation.  For  instance,  the  kinetic  energy  of  a mass  m 
with  linear  velocity  v is  mv2/2.  Its  kinetic  energy  of  rotation 
with  an  angular  velocity  co  is  loo2/ 2. 

If  the  whole  mass  of  the  rotating  body  be  considered 
concentrated  at  a distance  k from  the  axis  of  rotation,  such 
that  its  kinetic  energy  is  unaltered,  since  the  linear  velocity 
at  this  distance  is  v = k,  we  may  write 

mv2/ 2 = mk2  or/2  = 7co2/2 

Hence 

I = mk2 

and  k is  called  the  radius  of  gyration  of  the  body. 

It  may  be  shown  that  when  a body  of  mass  m has  a mo- 
ment of  inertia  IQ  about  an  axis  through  its  center  of  mass, 
its  moment  of  inertia  about  a parallel  axis  at  a distance  d 
from  the  first  is 

7 = 70  + md2 

The  method  of  finding  the  moment  of  inertia  of  the  bar 
is  to  set  it  in  rotation  by  a known  falling  mass.  Its  kinetic 
energy  of  rotation  is  then  the  difference  between  the  poten- 
tial energy  of  the  suspended  mass  and  its  kinetic  energy  at 
the  end  of  its  drop.  The  angular  velocity  having  been  de- 
termined, the  moment  of  inertia  is  calculated  from  the  kine- 
tic energy  of  rotation. 


92 


Manual  of  Experiments 


The  apparatus  consists  of  an  aluminum  disk  mounted 
on  a vertical  spindle,  and  a vertical  support  upon  which  are 
fixed  two  pulleys.  See  Fig.  25.  A known  mass  is  suspend- 
ed by  means  of  a cord  which  passes  over  the  upper  pulley 
and  under  the  lower  one,  and  is  wound  several  times  around 
a wooden  spool  on  the  vertical  spindle,  so  that  when  the 
mass  drops,  it  sets  the  disk  in  rotation.  The  metal  bar 
whose  moment  of  inertia  is  to  be  found  is  placed  on  the  disk 
and  held  in  position  by  projecting  pins  which  fit  into  holes 


in  the  disk.  The  disk  is  released  by  tilting  a small  lever 
which  catches  on  a projecting  pin  on  the  wooden  spool.  A 
small  adjustable  platform  on  the  upright  support  serves  to 
locate  the  height  to  which  the  mass  has  dropped  after  falling 
a given  number  of  seconds. 

As  the  mass  falls  its  potential  energy  is  converted  into 
kinetic  energy,  and  at  any  instant  its  loss  in  potential  energy, 
is  equal  to  its  gain  in  kinetic  energy  plus  that  acquired  by 
the  rotating  disc.  If  m be  the  falling  mass,  h the  distance 
it  falls  in  time  t,  g the  acceleration  of  gravity,  v the  final 


In  General  Physics 


93 


velocity  of  m at  the  end  of  its  fall,  co  the  angular  velocity  of 
the  disk  corresponding  to  v,  and  I the  moment  of  inertia  of 
the  rotating  disk,  spindle,  and  spool, 

m g h = }/%  m v2  2 / co2 

If  we  assume  the  friction  of  the  apparatus  to  be  constant 
the  mass  will  fall  with  constant  acceleration  a and  the  dis- 
tance fallen  will  be 

h = at2/ 2 

The  velocity  at  the  end  of  the  fall  will  be 

v = at. 

Hence  from  the  preceding  equation 

v = 2 h/t. 

The  linear  velocity  of  the  circumference  of  the  spool  will  at 
any  instant  be  the  same  as  that  of  the  falling  mass  since  they 
are  connected  by  a cord,  and  as  the  angular  velocity  of  the 
spool  equals  the  linear  velocity  of  its  circumference  divided 
by  its  radius  r}  the  angular  velocity  at  the  end  of  the  fall 
will  be 

a>  = v/r. 

' 

The  linear  and  angular  velocities  can  therefore  be  determined 
from  measurements  of  m,  h , t and  r and  the  value  of  / may 
be  calculated  from  the  equation  of  energy. 

When  the  bar  is  placed  on  the  disk  and  the  latter  is  set 
in  rotation  by  th£  larger  mass  mh  the  distance  fallen  hh  in 
the  same  time  t,  will  be  different,  and  therefore  the  velocities 
vi  and  coi  will  be  different.  If  A be  the  moment  of  inertia 
of  the  bar,  the  equation  of  energy  becomes 


pxus-^S-  Q- 
sl  —>  t ^ 

^ 17  <2  Y^ 


'g  fa  (yoJTTbi  / , 
/.  o.  n 


AS  v/<?'"-r- 


9' 


Manual  of  Experiments 


A i ^ «i  g h = -1/2  mi  A*  + H (/  + A)  coi2. 

_fa  ^ 

arid  wi  can  be  determined  as  before  and  / being 
known,  I\  can  be  calculated  from  the  equation. 

It  may  be  shown  that  the  moment  of  inertia  of  a rectan- 
gular bar  of  length  11 1”  width  “b”  and  mass  (i m ” is 


h = m (/2  + b 2)  /12. 


/i  can  be  calculated  from  this  equation  as  a check  on  the  . 
above  determination. 

Directions. — Loop  the  thread  over  the  projecting  pin 
on  the  spool  and  wrap  it  several  times  around  the  spool; 
then  pass  it  under  and  over  the  pulleys  as  shown  in  the  dia- 
gram and  suspend  the  mass  of  45  grams  from  the  other  end. 
Adjust  the  apparatus  until  the  bottom  of  the  mass  hangs 
level  with  the  mark  on  the  support  just  below  the  upper  pul- 
ley, and  catch  the  pin  on  the  lever  so  that  the  disk  does  not 
rotate.  Note  the  ticks  of  the  seconds  clock  and  when  all  is 
ready  tip  the  lever  and  release  the  disk  on  the  tick  of  the 
clock  sounder.  Watch  the  falling  mass  and  at  the  same 
time  count  the  ticks  of  the  sounder,  beginning  with  zero  at 
the  instant  of  release.  Observe  the  point  reached  by  the 
mass  on  say  the  seventh  tick,  or  at  the  end  of  the  seventh 
second  after  beginning  to  fall.  Repeat,  and  adjust  the 
movable  platform  until  the  mass  just  strikes  the  platform 
on  the  seventh  tick.  Measure  with  beam  compass  and  meter 
sticks  the  height  h through  which  the  mass  falls  in  seven 
seconds.  Find  also  the  radius  of  the  spool  by  allowing  the 
cord  to  unwind  slowly  from  the  spool  and  measuring  the 
distance  moved  through  by  the  suspended  mass  when  the 
spool  makes  five  complete  revolutions.  This  distance  will 
be  10  7r  times  the  radius  of  the  spool.  Calculate  v and  co 
and  from  the  equation  of  energy  obtain  the  value  of  /,  as- 


In  General  Physics 


95 


suming  g = 980  cm/sec2.  Then  place  the  steel  bar  on  the 
disk,  suspend  the  mass  of  100  grams  from  the  cord  and  re- 
peat the  above  performance.  From  the  data  obtained  and 
the  equation  of  energy  calculate  the  value  of  (/+/ 1)  and 
subtract  the  value  of  / previously  obtained. 

Next  measure  the  length  and  width  of  the  bar,  and  find 
its  mass  on  the  coarse  balance.  From  the  equation  given 
above  calculate  its  moment  of  inertia  and  compare  it  with 
the  value  of  I\  obtained  in  the  experiment. 

Finally  determine  by  about  ten  trials,  through  what  dis- 
tance you  can  raise  and  lower  the  platform  on  the  support 
without  being  able  to  detect  any  difference  between  the 
simultaneous  click  of  the  clock  sounder  and  that  of  the  fal- 
ing  mass  on  the  platform.  A table  of  formulas  for  various 
moments  of  inertia  will  be  found  in  the  appendix. 

Question: — a)  What  would  be  the  percent  error  in 
your  determination  of  / if  the  center  of  mass  of  the  bar  were 
displaced  one  mm.  from  the  axis  of  rotation  of  the  disk? 

What  percent  error  is  introduced  in  your  result  by  the 
possible  variation  in  the  height  h as  determined  in  your 
final  trials? 

16.  THE  TORSION  PENDULUM. 

To  show  the  relation  between  the  period  of  a torsion  pen- 
dulum and  the  length  and  radius  of  its  suspension;  to  find 
the  moment  of  inertia  of  a ring  and  of  a cylinder  by  the 
torsion  pendulum,  and  the  modulus  of  rigidity  of  the 
suspending  wire. 

(D.  81-89,  118,  119,  115;  G.  89A,  736;  K.  125,  135-140;  W. 
85,  175.) 

Apparatus. — Pendulum  support,  disk,  ring,  beam,  com- 
pass, meter  stick,  coarse  balance,  weights,  micrometer  cal- 


96  Manual  of  Experiments 

ipers,  two  cylinders,  and  three  wires,  two  of  the  same  length 
but  different  diameters,  the  third  of  a different  length 
with  the  same  diameter  as  one  of  the  other  two. 

Theory,  Method,  and  Description. — (See  also  theory 
of  experiments  14  and  15.)  The  torsion  pendulum  consists 
of  a uniform,  vertical,  straight  wire  clamped  at  one  end  and 
supporting  a heavy  body  at  the  other  end.  If  the  wire  be 
twisted  through  a small  angle  without  displacing  it,  and  then 
released,  the  body  will  pass  its  position  of  rest,  owing  to  its 
kinetic  energy  and  will  perform  angular  vibrations.  By 
Hook’s  law  the  restoring  couple  at  any  instant  will  be  pro- 
portional to  the  twist  of  the  wire,  and  the  vibrations  will  be 
harmonic  and  isochronous.  The  period  of  the  pendulum 
is  the  time  required  to  make  a complete  vibration,  or  double 
swing;  that  is,  the  time  which  elapses  between  two  successive 
passages  of  the  pendulum  in  the  same  direction  through 
some  point  of  its  path,  such  as  its  position  of  rest. 

It  may  be  shown  that  the  period  of  such  a pendulum  is 

T = 2 7r  V I / M i 

where  / is  its  moment  of  inertia  and  Mi  the  moment  of  the 
torsional  couple  when  the  wire  is  twisted  through  an  angle 
of  one  radian.  But  Mi  depends  upon  the  material,  radius 
and  length  of  the  wire.  . In  experiment  14  the  value  of  the 
modulus  of  rigidity  was  given  as  n — 21M / —2lMi/ 
(71T4)  where  l is  the  length  and  r the  radius  of  the  wire.  From 
this  we  get  Mi  = 7mr4/2/  and  substituting  this  value  of  Mi 
in  our  expression  for  the  period,  we  have 

T = 2 7r  V /T~(277rnr47  = 2 tt  V 1 1 C 

where  1/C  is  a constant  for  a given  wire  (the  constant  of 
torsion) . 


In  General  Physics 


97 


From  this  equation  we  see  that  the  period  is  directly  pro- 
portional to  the  square  root  of  the  moment  of  inertia  of  the 
body,  and  to  the  square  root  of  the  length  of  the  wire,  and 
inversely  as  the  square  of  its  radius. 

• By  comparing  the  periods  of  two  pendulums  of  the  same 
wire  with  the  same  body  suspended  but  having  different 
lengths,  we  can  show  that  the  second  of  these  statements  is 
true;  by  comparing  one  of  these  periods  with  that  of  a pen- 
dulum of  the  same  length  and  suspended  mass,  but  with  a 
wire  of  different  diameter  we  may  show  that  the  third  state- 
ment is  true.  The  first  statement  offers  a means  of  deter- 
mining the  moment  of  inertia  of  the  body. 

To  do  this  we  use  a pendulum  consisting  of  a heavy  frame 
similar  to  that  in  Fig.  21,  at  the  top  of  which  is  clamped  a 
wire  supporting  a heavy  iron  disk.  A pointed  stop  may  be 
pressed  into  contact  with  the  center  of  the  disk  and  prevents 
oscillations  other  than  those  about  this  center.  The  period 
of  this  pendulum  is  observed  and  will  be 

T — 2 7r  V 7 / Mi 

where  7 and  Mi  are  respectively  the  moment  of  inertia  of 
the  pendulum  and  moment  of  the  torsional  couple  for  unit 
angle.  The  body  whose  moment  of  inertia  is  to  be  deter- 
mined is  then  fastened  to  the  pendulum  disk  so  that  it  is 
symmetrically  placed  with  respect  to  the  wire  and  disk. 
The  period  of  this  pendulum  is  then  determined  and  will  be 

T\  — 2 7r  V (7  + 1\)  / Mi 

< 

where  Ii  is  the  moment  of  inertia  of  the  body.  From  these 
two  equations  we  get  the  proportion 

T 2 : Ti 2 ::/:(/  + h) 

£ 

from  which  we  calculate  7i  in  terms  of  the  known  moment  7 
and  the  two  periods. 


98 


Manual  of  Experiments 


The  moment  of  inertia  of  the  disk  about  an  axis  perpen- 
dicular to  its  face  is  Y2  mP  where  m is  its  mass  and  r its  radius. 

We  may  also  calculate  A from  the  dimensions  of  the  body 
as  a check  on  the  value  obtained  above.  For  instance  if 
the  body  is  a ring  of  radii  rh  and  r2  and  has  a mass  m, 

1 1 = m ( ri  + r2)/ 2 

(See  Appendix  for  formulae  of  moments  of  inertia.) 

From  the  expression  for  the  period  of  the  pendulum  we 
may  determine  the  modulus  of  rigidity  of  the  suspending 
wire,  when  the  other  factors  are  known. 

Directions: — Preliminary. — Measure  the  mass  of  disk 
and  ring  on  the  coarse  balance,  and  their  radii  with  the  meter 
stick.  Measure  also  the  radii  of  the  wires  with  the  micro- 
meter calipers.  Their  lengths  are  best  measured  with  beam 
compass  and  meter  stick  when  they  are  suspended  and 
stretched. 

Moment  of  inertia  of  ring. — Pass  the  shorter  wire  through 
the  disk  and  clamp  it  in  the  support.  Measure  its  length. 
Press  the  stop  into  contact  with  the  center  of  the  disk,  dis- 
place the  latter  through  a small  angle,  and  allow  it  to  os- 
cillate. When  the  oscillations  are  quite  steady  release  the 
stop,  and  count  the  time  for  ten  complete  oscillations,  by 
noting  the  passage  of  the  index  mark  on  the  disk  past  the 
pointer  on  the  frame.  Repeat  as  a check  and  calculate  the 
period  of  the  pendulum.  Unclamp  the  wire,  slip  the  ring 
over  it  and  replace  it.  The  ring  must  be  placed  symmetri- 
cally on  the  disk  with  respect  to  its  axis.  Determine  the 
period  of  this  pendulum  as  before.  Calculate  the  moment 
of  inertia  of  the  disk  from  its  mass  and  radius  and  that  of 
the  ring  from  the  periods  and  moment  of  inertia  of  the  disk. 
Calculate  its  moment  of  inertia  from  its  mass  and  radii  also. 
Then  compare  these  two  moments  of  inertia  for  the  ring. 


In  General  Physics 


99 


Moment  of  inertia  of  two  equal  cylinders  about  an  axis 
parallel  to  their  own. — Two  cylinders  of  equal  mass  and  di- 
mensions are  placed  on  the  disk  of  the  pendulum  at  opposite 
ends  of  a diameter.  If  I be  the  moment  of  inertia  of  the 
disk  alone  about  an  axis  through  its  center  (the  suspending 
wire),  1 1 the  moment  of  inertia  of  one  cylinder  about  its 
own  axis  and  /2  the  moment  of  inertia  of  one  cylinder  about 
the  axis  of  the  disk  we  may  determine  / from  the  dimensions 
of  the  disk  mr2.  Then  by  determining  the  period  T of 
the  pendulum  alone  and  the  period  7\  of  the  pendulum  and 
two  cylinders  we  have  T2  : 7\2  = / : / + 2/2.  Hence  we 
may  calculate  /2.  But  if  mx  be  the  mass  of  one  cylinder 
and  r2  the  distance  between  its  axis  and  the  axis  of  rotation 
we  have 

/ 1 = 1 2 — m-x  r22 

(See  Exp.  15  Theory).  Hence  we  may  calculate  I\.  As  a 
check  / 1 may  be  calculated  by  the  formula 

I\  = Vi  mn2 

where  wi  is  the  mass  and  ri  the  radius  of  each  cylinder. 
We  have  therefore  to  determine  the  period  of  the  pendulum 
alone  and  that  of  pendulum  and  cylinders  together,  the 
mass  of  the  disk  and  its  radius,  the  mass  of  each  cylinder 
and  its  radius  and  the  distance  between  their  axes  in  order 
to  determine  1 1. 

Variation  of  period  with  length  and  with  radius  of  sus- 
pension.— Remove  the  ring  and  replace  the  wire  suspension 
by  the  longer  one  of  the  same  diameter,  and  determine  its 
period  as  before.  Then  replace  this  wire  by  the  third  one 
of  the  same  length  but  of  different  diameter,  and  determine 
its  period.  Show  how  the  period  varies  with  the  length  of 
suspension  by  calculating  the  ratio  of  period  to  square  root 


100 


Manual  of  Experiments 


of  length  in  the  two  cases  where  the  diameter  and  mass  were 
the  same  but  the  length  varied.  Show  how  the  period  varies 
with  radius  of  suspension  by  calculating  the  products  of 
period  and  square  of  radius  in  the  two  cases  where  the  length 
and  mass  were  constant  but  the  diameter  varied. 

Modulus  of  Rigidity  of  Steel  {or  Brass.) — From  the  first 
period  determined  calculate  the  modulus  of  rigidity  of  the 
suspension  wire. 

Question. — Which  of  the  measurements  in  this  experi- 
ment must  be  made  with  the  greatest  care,  and  why? 

J 6<7-M 

17.  THE  SPIRAL  SPRING  BALANCE 

To  calibrate  a spiral  spring  balance;  to  test  Hook’s 
law  for  a spiral  spring;  and  to  determine  the  weight  and 
specific  gravity  of  several  substances  with  the  spiral  spring 
balance.  (D.  162,  168,  191.  G.  27,  112,  118,  119,  122. 
K.  176,  182,  183.  W.  127,  129). 

Apparatus. — Spiral  spring  balance,  set  of  weights,  (1 
to  5 grams)  forceps,  samples  of  lead,  brass  and  aluminum, 
small  beaker  for  distilled  water,  small  beaker  of  kerosene  or 
alcohol. 

Method,  Theory  and  Description. — The  specific  grav- 
ity of  a substance  is  the  ratio  of  its  density  to  that  of  some  stand- 
ard substance.  The  standard  usually  employed  is  water  at 
4°  C.  Since  the  density  of  a substance  is  its  mass  per  unit 
volume,  it  follows  that  the  specific  gravity  of  a substance 
may  be  defined  as  the  ratio  of  its  mass  to  that  of  an  equal 
volume  of  water.  The  method  employed  is  based  upon 
Archimedes’  principal  which  states  that  a body  immersed  in 
a fluid  is  buoyed  up  by  a force  which  is  equal  {and  opposite) 
to  the  weight  of  the  volume  of  the  fluid  which  the  body  displaces. 


J 

t 

■ 


In  General  Physics 


101 


In  the  spiral  spring  balance  to  be  used,  we  have  a spiral 
spring  to  which  is  suspended  a small  pan  to  hold  the  sub- 
stance under  experiment.  See  Fig.  26.  The  spring  S 


a 


is  supported  by  two  vertical  telescoping  tubes  T mounted  on 
a tripod  base  with  leveling  screws.  It  is  suspended  from 
the  inner  tube  which  is  graduated  in  millimeters  for  about 
half  a meter  of  its  length;  and  can  be  moved  up  or  down 


102 


Manual  of  Experiments 


by  means  of  a chain  rack  and  pinion  R near  the  base.  Be- 
tween the  spring  and  pan  is  an  indicator  / consisting  of  a 
small  aluminum  rod  with  two  cross  arms,  one  at  either  end 
of  a short  glass  tube  enclosing  the  rod  and  supported  by 
an  adjustable  clamp  on  the  outer  tube.  (See  Fig.  26a.) 
The  glass  is  whitened  at  the  back,  and  has  a fine  black 
line  passing  entirely  around  its  middle.  The  aluminum 
rod  carries  at  its  center  a small  cylinder  bearing  three 
equidistant  black  lines,  the  middle  one  of  which  is  made  to 
coincide  with  the  line  on  the  glass  tube  when  the  spring  is 
brought  to  the  zero  position.  A vernier  V on  the  upper 
end  of  the  tube,  reads  to  tenths  of  a millimeter  and  gives 
the  elongation  of  the  spring  under  any  load  when  the  in- 
dicator is  in  the  zero  position.  On  the  outer  tube  is  also 
an  adjustable  support  P for  a vessel  with  liquid.  According 
to  Hook’s  law  the  stretch  of  the  spring  is  (within  limits) 
directly  proportional  to  the  force  (weight)  producing  the 
stretch. 

To  determine  the  weight  of  an  object  with  the  balance, 
the  latter  must  first  be  calibrated.  That  is,  a curve  must  be 
drawn  through  points  obtained  by  plotting  a number  of 
known  weights  and  the  corresponding  stretch  when  each  is 
placed  in  the  pan.  Upon  observing  the  stretch  produced 
by  any  object  in  the  pan,  the  corresponding  weight  can  be 
read  off  directly  from  the  curve. 

To  determine  the  specific  gravity  of  a solid  (not  soluble 
in  water)  we  first  determine  its  weight  when  suspended  in 
air,  W±,  then  when  it  is  completely  immersed  in  water,  W%. 
The  weight  of  an  equal  volume  of  water  is  then,  by  Archi- 
medes ’ principle,  equal  to  their  difference  ( W\  — W 2)  and 
the  expression  for  the  specific  gravity  of  the  body  is 

sp.  gr.  = WJ  (W1-W2). 


In  General  Physics 


103 


We  are  not  interested  however  in  the  actual  weight  of  the 
body,  but  in  its  ratio  to  that  of  an  equal  volume  of  water, 
and  since  the  stretch  of  the  spring  is  directly  proportional 
to  the  weight  producing  it  and  we  are  here  dealing  only  with 
ratios  and  differences  of  weights,  we  can  make  our  calcula- 
tions with  the  stretches  themselves.  That  is  since  the  weight 
in  each  case  is  some  constant  K times  the  corresponding 
stretch,  W = KS  and 

KSi  Si 

sp.  gr.  = - — = — 

KS1  — KS2  S1-S2 

where  Si  and  S2  are  the  stretches  corresponding  to  W 1 and  W2. 

To  determine  the  specific  gravity  of  a liquid,  a solid  which 
has  a greater  density  than  the  liquid  is  weighed  first  when 
suspended  in  air,  then  when  completely  immersed  in  water, 
and  finally  when  immersed  to  the  same  depth  in  the  liquid 
under  experiment. 

Directions. — To  draw  the  calibration  curve  and  test 
Hook’s  law. — Adjust  the  leveling  screws  and  glass  tube  so 
that  the  indicator  hangs  in  the  center  of  the  latter.  Place 
the  eye  in  the  plane  of  the  circle  on  the  glass  tube  and  turn 
the  milled  head  which  moves  the  inner  tube  until  the  middle 
mark  of  the  indicator  is  in  this  plane.  Observe  the  reading 
on  the  vernier  and  record  this  no  load  reading  R0. 

Place  a one  gram  weight  in  the  pan  (using  the  forceps  to 
handle  it)  and  turn  the  milled  head  until  the  indicator  is  in 
its  zero  position  as  before.  Observe  and  record  the  load 
reading  R..  Do  the  same  for  2,  3,  4,  and  5 grams,  and  then 
repeat  the  no  load  reading  as  a check. 

Calculate  the  stretch  R — R0  for  each  weight.  Plot  the 
known  weights  as  abscissae,  and  the  corresponding  stretches 
as  ordinates  and  draw  the  calibration  curve  through  the 


104 


Manual  of  Experiments 


points  obtained.  If  this  curve  is  a straight  line  the  spring 
obeys  Hook’s  law.  (See  introduction  for  method  of  plotting 
curves). 

To  find  the  weight  of  the  samples  of  lead , brass , and  alum- 
inum.— Place  each  object  in  the  pan  and  record  the  resulting 
load  reading.  Also  record  the  no  load  reading  after  remov- 
ing each  object,  and  calculate  the  stretch  produced  by  each. 
Refer  the  stretch  to  the  calibration  curve  and  note  from  the 
axis  of  abscissae  the  corresponding  weight  of  the  object. 

To  find  the  specific  gravity  of  the  objects. — Suspend  about 
three  inches  of  the  fine  wire  from  the  pan  and  record  the  re- 
sulting reading  i?0.  Place  the  object  in  the  pan  and  record 
the  resulting  reading  R\.  Calculate  the  stretch  Si  = Ri  — Ro. 

Remove  the  object,  fill  the  beaker  with  distilled  water 
and  place  it  on  the  support.  Adjust  this  support  until  the 
wire  is  immersed  in  the  water  except  for  about  a centimeter 
of  its  upper  end.  Record  the  resulting  reading  i?2.  To 
make  the  effect  of  surface  tension  as  small  as  possible,  have 
only  a single  strand  of  the  wire  passing  through  the  surface 
of  the  water.  Then  fasten  the  wire  securely  to  the  object 
and  immerse  them  to  the  same  point  on  the  wire.  To  avoid 
errors  take  care  that  no  air  bubbles  adhere  to  the  wire  or  to 
the  sample  when  they  are  immersed  in  the  water.  Record 
the  reading  i?3  and  calculate  the  corresponding  stretch  S2. 
Since  the  effect  of  surface  tension  is  included  in  both  read- 
ings i?2  and  i?3,  it  is  eliminated  from  their  difference  S2. 
From  Si  and  S2  calculate  the  corresponding  specific  gravity 
of  the  object.  Do  the  same  for  each  of  the  three  samples. 

To  find  the  specific  gravity  of  kerosene  or  alcohol ., — Suspend 
the  wire  in  the  liquid  as  before  and  record  the  resulting  read- 
ing i?4.  Then  suspend  the  brass  sample  at  the  same  depth 
as  it  was  in  the  water,  and  record  the  reading  i?5.  Calcu- 
late the  corresponding  stretch  S3.  Note  the  previous 


In  General  Physics 


105 


♦ 

stretches  for  brass,  Si  and  S2  and  calculate  the  specific  grav- 
ity of  the  liquid  from  these  three  stretches. 

Question  {a). — Derive  the  expression  for  calculating  the 
specific  gravity  of  the  liquid  from  the  data  taken,  (b). 
Compare  density  and  specific  gravity  of  a body  in  the  C.G.S. 
system;  in  the  F.P.S.  system. 

18.  SPECIFIC  GRAVITY  WITH  HYDROMETERS. 


(a)  To  determine  the  specific  gravity  of  solids  with  a 
Nicholson’s  hydrometer,  (b)  To  determine  the  specific 
gravity  of  a liquid  with  a Beaume  hydrometer.  (D.  162; 
G.  121-129;  K.  182,  185;  W.  147.) 

Apparatus. — Nicholson’s  hydrometer,  tall  hydrometer 
jar,  set  known  masses,  two  metal  pieces,  Beaume  hydro- 
meter, jars  of  solutions. 

Description  and  Theory.— A Nicholson’s  hydrometer 
consists  of  a cylindrical  metal  vessel  B at  one  end  of  which 
is  attached  a loaded  chamber  D (Fig.  27  b.)  The  latter 
serves  to  keep  the  vessel  in  an  upright  position  when  placed 
in  a liquid.  A slender  stem  S,  carrying  a small  pan  for 
holding  objects,  projects  above  the  metal  chamber. 

Let  Mi  be  the  mass  which,  when  placed  on  the  pan  P, 
will  cause  the  hydrometer  to  sink  in  water  to  a given  mark 
0 on  the  stem  S;  M2  the  mass  in  addition  to  the  object,  whose 
specific  gravity  is  desired,  to  sink  the  vessel  to  the  same 
mark;  and  M3  the  mass  to  produce  the  same  result  when  the 
object  N is  placed  in  the  pan  D in  the  water.  Then  M\-M2 
is  the  mass  of  the  object  in  air  and  M3-M2  is  the  apparent 
loss  of  mass  in  water.  Since  the  specific  gravity  is  equal*  to 


Manual  of  Experiments 


♦ 


106 


the  mass  of  the  body  in  air  divided  by  the  apparent  loss  of 
mass  in  water  we  have 

M1-M2. 

Specific  gravity  = 

M$-M2 

The  Beaume  hydrometer  is  of  the  variable  immersion 


Fig.  27  a and  b 

type  and  consists  of  a glass  bulb  A (Fig.  27  a)  with  an  elon- 
gated stem.  The  bulb  is  loaded  at  one  end  so  as  to  make 
it  float  in  an  upright  position.  The  elongated  stem  has  a 
scale  which  has  been  graduated  so  that  its  reading  at  the 


In  General  Physics 


107 


surface  of  the  liquid  in  which  the  hydrometer  has  been  placed 
gives  the  specific  gravity  of  the  liquid.  Care  must  be  taken 
in  reading  the  hydrometer  to  avoid  errors  due  to  the  distor- 
tion of  the  surface  around  the  stem  by  capillary  action. 
Thus  in  Fig.  27a  the  reading  C is  in  line  with  the  under  sur- 
face of  the  liquid. 


Directions. — Determination  of  the  specific  gravity  of  a 
piece  of  metal  with  Nicholson’s  hydrometer. — Pour  distilled 
water  into  the  hydrometer  jar  until  there  is  enough  to  float 
the  hydrometer.  When  placing  the  hydrometer  into  the  jar 
see  that  no  air  bubbles  cling  to  the  vessel.  Place  the  card- 
board E around  the  stem  in  order  to  prevent  the  hydrometer 
from  sinking  if  the  mass  added  to  the  pan  is  too  large. 
Place  known  masses  upon  the  pan  P till  the  mark  on  the 
stem  S is  just  at  the  surface  of  the  water  and  record  the 
value  of  these  masses  as  Mi.  Remove  the  known  masses 
from  the  pan  and  place  in  it  one  of  the  pieces  of  metal. 
Again  add  known  masses  till  they,  together  with  the  piece  of 
metal,  sink  the  hydrometer  to  the  mark  on  the  stem.  Call 
the  value  of  the  known  masses  in  this  case  M2.  Put  the 
piece  of  metal  in  the  lower  pan  D and  determine  the  mass 
Ms  which  must  be  placed  in  the  pan  P to  sink  the  hydro- 
meter to  the  mark  on  the  stem.  From  the  data  obtained 
calculate  the  specific  gravity  of  the  metal  by  means  of  the 
relation 


Specific  gravity  = 


Mi~  M2 


A/3-A/2 


Repeat  the  above  determination  and  make  a second  cal- 
culation of  the  specific  gravity  of  the  metal  as  a check  on 
the  first.  In  a similar  manner  make  two  determinations 
of  the  specific  gravity  of  each  of  the  metals  furnished  you. 


108 


Manual  of  Experiments 


Specific  gravity  with  the  Beaume  hydrometer . — Liquids 
whose  specific  gravity  are  to  be  determined  will  be  found 
in  jars  near  the  sink.  Place  the  hydrometer  in  each  solu- 
tion and  note  the  mark  on  the  stem  which  is  at  the  surface 
of  the  liquid.  This  mark  C , Fig.  27a,  can  be  read  most  ad- 
vantageous^ by  sighting  along  the  under  surface  of  the  li- 
quid through  the  walls  of  the  jar.  After  removing  the  hy- 
drometer from  one  solution,  wash  and  dry  thoroughly  before 
placing  it  in  another  solution.  Make  at  least  three  separate 
readings  for  the  specific  gravity  of  each  liquid. 

Questions. — 1.  What  effect  does  capillarity  have  upon 
the  reading  for  the  specific  gravity  of  a liquid  with  the 
Beaume  hydrometer?  2.  In  the  determination  of  the 
specific  gravity  of  a metal  with  Nicholson’s  hydrometer 
what  will  be  the  effect  on  the  result  of  trapping  a bubble 
of  air  under  the  metal  object  when  it  is  placed  in  the  pan  Z)? 


19.  SPECIFIC  GRAVITY  BY  HARE’S  METHOD. 

To  determine  the  specific  gravitv  of  liquids.  (K.  174, 
182-185;  W.  144.) 

Apparatus. — Hare  's  apparatus,  beakers,  distilled  water, 
meter  stick,  measuring  rod,  liquids. 

Description  and  Theory. — The  Hare's  apparatus  used 
for  finding  the  specific  gravity  of  liquids  consists  of  three 
long  glass  tubes  fastened  in  a vertical  position  on  a common 
base.  The  lower  ends  of  the  tubes  dip  into  the  liquids  whose 
specific  gravity  is  to  be  determined  and  the  upper  ends  are 

joined  together.  If  the  air  is  drawn  out  from  the  tubes  the 

# 

liquids  will  rise  in  them  due  to  the  difference  between  the 


In  General  Physics 


109 

a 

pressures  on  the  inside  and  the  outside.  The  pressure  on 
the  outside  is  the  same  for  each  tube,  namely  one  atmosphere, 
and  since  the  tubes  communicate  with  each  other,  the  pres- 
sure on  the  inside  must  be  the  same.  The  pressure  on  the 
inside  is  made  up  of  the  pressure  of  the  column  of  liquid  and 


the  pressure  of  the  remaining  air.  The  pressure  per  unit 
area  exerted  by  a column  of  liquid  depends  only  upon  the 
height  of  the  column  and  the  density  of  the  liquid.  If  the 
heights  to  which  the  liquids  rise  in  the  tubes  are  represented 
by  a,b , and  c,  (Fig.  28)  and  their  respective  specific  gravities 


110 


Manual  of  Experiments 


by  di,  d2  and  d%,  the  quantities  dxa , d2b  and  d^c,  are  pro- 
portional to  the  pressures  per  unit  area  at  the  bottom  of 
each  tube  due  to  the  liquids.  But  this  pressure  is  the  same 
for  each  tube  and  hence 

d\U  = d2b  = d$c  . . - . (1) 

If  one  of  the  liquids,  say  the  second,  is  water,  its  specific 
gravity  ^2  = 1,  and  the  specific  gravity  of  the  other  liquids 
can  be  obtained  from  the  relations  di  = ~a  and  d%  = ~,  where 
a,  b,  and  c are  the  heights  of  the  columns  of  liquids  which 
can  be  determined  by  direct  measurement. 

Directions. — With  the  beakers  filled  with  the  liquids 
whose  specific  gravity  is  to  be  determined,  decrease  the  pres- 
sure in  the  tubes  by  means  of  suction  till  the  lighter  liquid 
has  risen  almost  to  the  top  of  the  tube  containing  it.  Close 
the  pinch  cock  B on  the  rubber  tube  so  as  to  keep  the  liquids 
at  constant  heights  in  the  tubes.  By  means  of  the  measur- 
ing rod  and  meter  stick  determine  the  height  of  the  liquid  in 
each  tube  above  the  level  of  the  liquid  in  its  respective  beak- 
er. Designate  the  height  of  the  lighter  liquid,  the  water 
and  the  heavier  liquid,  by  a , b and  c,  respectively,  and  let 
di,  d2  dz  be  the  corresponding  specific  gravities.  If  d2  is 
taken  equal  to  unity,  the  specific  gravity  of  the  lighter  liquid 
is  found  from  the  relation  di  = -a  and  that  of  the  heavier 
liquid  from  ds  = \ 

Change  the  pressure  slightly  in  the  tubes  so  as  to  cause 
the  heights  of  the  liquid  columns  to  be  different  from  what 
they  were  before  and  again  determine  the  specific  gravities 
of  the  liquids  from  measurements  of  the  heights  of  the  col- 
umns. Repeat  the  above,  varying  slightly  the  heights  of  the 
liquids  each  time,  till  five  determinations  of  the  specific 
gravities  of  the  liquids  have  been  made.  Record  the  mean 
of  the  five  determinations  for  each  liquid. 


In  General  Physics 


111 


According  to  the  directions  on  the  use  of  the  Beaume 
hydrometer  for  determinating  the  specific  gravity  of  liquids 
given  under  experiment  18,  note  the  readings  of  the  hydrome- 
ter in  the  vessels  near  the  sink  containing  the  lighter  and 
the  heavier  liquids.  Use  great  care  in  rinsing  and  drying 
the  hydrometer  before  placing  it  in  either  solution.  Com- 
pare the  readings  of  the  hydrometer  with  the  values  ob- 
tained above. 

Questions. — 1.  What  objection  is  there  to  having  the 
vertical  tubes  of  unequal  diameter?  2.  Assuming  b to  be 
correct  what  percent  error  in  d%  will  an  error  of  2 mm.  in 
measurement  of  c produce,  (a)  in  your  first  determination, 
(b)  in  your  last  determination? 


20.  SURFACE  TENSION. 

(a)  To  determine  the  value  of  the  surface  tension  of 
water  by  capillary  tubes,  (b)  To  determine  the  value  of 
the  surface  tension  of  water  by  means  of  the  Jolly  balance. 
(D.  207-211;  G.  138;  K.  254-257,  263;  W.  156-159.) 

Part  1.  Surface  Tension  by  Capillary  Tubes. 

Apparatus. — Capillary  tube,  stand  and  scale  for  holding 
tube,  beaker  of  distilled  water,  mercury,  fine  balance. 

Description  and  Theory. — The  surface  of  a liquid  is 
in  a state  of  tension  similar  to  the  tension  of  a tightly 
stretched  drum  head.  This  tension  is  manifested  by  the 
tendency  of  the  surface  to  contract  so  as  to  enclose  as  small 
a volume  as  possible,  as  is  seen  in  a rain  drop.  Imagine  a 
line  drawn  in  any  direction  across  the  surface  of  a liquid. 


112 


Manual  of  Experiments 


The  value  of  the  force  upon  unit  length  of  this  line  is  called 
the  “surface  tension’  ’ of  the  liquid.  By  virtue  of  this  force, 
a liquid  which  wets  a tube  will  rise  in  it  above  the  general 
surface  of  the  liquid  when  the  tube  is  placed  vertically  in 


the  liquid.  As  soon  as  the  liquid  in  the  tube  rises  above 
the  normal  surface,  gravity  tends  to  pull  it  downward.  The 
liquid  will  then  rise  in  the  tube  only  to  such  a height  that 
the  maximum  tension  which  the  surface  can  exert  is  just 
balanced  by  the  weight  of  the  liquid  in  the  tube. 

If  T is  the  surface  tension,  or  the  force  exerted  by  the 
surface  on  a line  one  centimeter  long,  and  r is  the  radius  of 
the  tube  (Fig.  29)  the  total  force  acting  around  the  circum- 
ference of  the  tube  is  27rr  T.  If  the  surface  of  the  liquid  makes 
the  angle  d with  the  walls  of  the  tube  at  the  point  of  contact, 
the  total  force  acting  upward  along  the  wall  of  the  tube  is 


In  General  Physics 


113 


2 7r  r T cos  6.  The  weight  of  the  liquid  which  this  force 
supports  is  7 r r2  h d g,  where  r is  the  radius  of  the  tube,  d 
the  density  of  the  liquid,  h the  height  to  which  the  liquid 
rises  and  g the  acceleration  of  gravity.  Then, 


2irr  T COS  0 = 77*  r2  hdg  ...  (1) 


from  which 


r h d g 
2 cos  6 


In  the  case  of  very  small  tubes,  the  cosine  of  the  angle 
of  contact  6 differs  little  from  unity.  Hence  (2)  may  be 
written  in  the  form 


T = 34  r h d g . . . (3) 

From  this  equation  it  is  seen  that,  if  the  height  h to  which 
a liquid  of  density  d rises  in  a tube  of  radius  r is  known,  the 
value  of  the  surface  tension  is  readily  obtained. 

Directions. — Take  a glass  tube  of  small  bore  and  clean 
it  carefully  with  a caustic  potash  solution,  then  rinse  it 
thoroughly  in  running  water.  After  cleaning  avoid  putting 
the  finger  over  the  end  of  the  tube  as  this  might  soil  the 
inner  surface.  To  measure  A,  fasten  the  tube  to  the  verti- 
cal scale  by  means  of  rubber  bands.  Place  the  scale  and 
tube  vertically  in  a beaker  of  distilled  water  with  the  base 
upon  which  the  scale  is  erected  resting  on  the  edge  of  the 
vessel.  Lower  the  tube  and  raise  it  again  to  be  sure  that  it 
is  wet  as  high  as  the  column  will  rise.  Note  the  position 
of  the  surface  of  the  liquid  on  the  scale  by  sighting  through 
the  walls  of  the  beaker  along  the  under  surface  of  the  liquid. 
Read  the  position  on  the  scale  to  which  the  water  rises  in 
the  tube.  From  the  readings  made  determine  the  height  A 
of  the  water  column  in  the  tube.  Move  the  tube  up  or 


114 


Manual  oe  Experiments 

down  a few  millimeters  and  repeat  the  above  measurements. 
Make  five  determinations  of  h and  take  the  mean  value. 

To  measure  r the  radius  of  the  capillary  tube,  see  that 
the  tube  is  perfectly  dry,  weigh  it  on  the  fine  balance,  and 
then  draw  into  it  a thread  of  mercury  about  3 or  4 cm.  in 
length  and  weigh  again.  The  difference  between  the  two 
weights  is  the  weight  of  the  mercury  in  the  tube.  Measure 
the  length  of  the  mercury  thread,  /.  The  weight  is 

m = d 7r  r2  / . . ,.  (4) 

and 

/ '.  m 

r = V 13.55  7T  / ...  (5) 

where  the  density  of  mercury,  d for  20°C  = 13,55. 

Substitute  in  equation  (3)  the  values  obtained,  and  de- 
termine the  value  of  the  surface  tension.  Repeat  the  above 
determination  with  a tube  of  different  diameter. 

Slight  impurities  on  the  surface,  such  as  an  impercept- 
ible quantity  of  oil  from  the  fingers,  may  change  the  value 
of  T very  much.  The  vessel  and  tube  should  therefore 
be  carefully  cleansed  and  rinsed  with  distilled  water.  The 
fingers  should  not  be  allowed  to  touch  the  water. 

Questions. — 1.  How  does  h vary  with  the  diameter 
of  the  tube?  2.  Would  this  method  be  applicable  to  li- 
quids which  do  not  wet  the  walls  of  the  tube?  Explain 
your  answer. 

Part  2.  Surface  Tension  by  Means  of  the  Jolly  Balance 

Apparatus. — Jolly  balance,  set  of  known  masses,  wire 
rectangle,  beaker  of  distilled  water. 

Description  and  Theory. — If  a wire  in  the  form  of  a 
rectangle  is  immersed  in  a liquid  and  then  raised,  it  is  ob- 
served that  the  liquid  clings  to  the  wire  in  the  form  of  a 


In  General  Physics 


115 


thin  film.  Suppose  a wire  in  the  form  of  a rectangle  is  sus- 
pended from  the  spring  of  a Jolly  balance  and  the  position 
of  the  scale  read  when  only  the  ends  of  the  rectangular  wire 
dip  in  the  liquid.  Then,  if  the  rectangle  is  immersed  in  the 
water  and  afterwards  the  spring  is  moved  up  till  the  film 
which  is  formed  just  breaks  and  the  position  of  the  scale 
read,  the  difference  between  the  readings  in  the  two  cases 
gives  a measure  of  the  pull  due  to  the  surface  tension.  If 
a represents  the  average  elongation  of  the  spring  per  gram 

and  e is  the  elongation  which  represents  the  pull  of  the  film 
e 

on  the  wire,  - g is  the  force  due  to  surface  tension,  g being 

the  force  on  one  gram  due  to  gravity.  If  the  rectangle  has 
a width  Wy  then 


t — JL  & 
a 2 W 


The  factor  2 is  introduced  because  the  film  has  two  sur- 
faces. 

Directions. — Place  known  masses  2,  3,  and  4 grams  on 
the  pan  of  the  balance  and  determine  the  elongation  a of 
the  spring  per  gram  added.  (See  experiment  17.)  Clean 
the  wire  rectangle  by  dipping  it  in  a caustic  potash  solution 
and  then  holding  it  in  running  water.  Handle  it  by  means 
of  the  fine  wire  attached  to  the  middle  of  the  frame.  Fasten 
the  wire  rectangle  to  the  hook  under  the  pan  and  lower  it 
into  a beaker  of  distilled  water.  Raise  the  spring  slowly 
and  note  approximately  the  depth  of  the  ends  of  the  rectangle 
below  the  surface  of  the  water  when  the  film  breaks.  Repeat 
this  two  or  three  times.  Now  lower  the  rectangle  till  the 
ends  of  the  wire  project  into  the  water  to  a depth  equal  to 
the  depth  found  above  and  observe  the  reading  on  the  scale. 
Call  this  the  zero  reading.  Again  lower  the  wire  into  the 
water  and  raise  the  spring  very  slowly  till  the  film  on  the 
wire  just  breaks.  Note  the  reading  of  the  vernier.  The 


116 


Manual  of  Experiments 


difference  between  this  reading  and  the  zero  reading  gives 
the  elongation  e of  the  spring  due  to  the  surface  tension 
of  the  film.  Measure  the  width  W.  (Fig.  30.)  of  the  wire 


SE.CTIOIN 
on  R E3 

5HOWINO 
THE.  Two 
Fu.M5. 


rectangle  and  substitute  the  values  determined  in  equation 
(1)  and  solve  for  T. 

Repeat  these  determinations  four  more  times  and  find 
the  mean  value  of  the  surface  tension  found  for  the  five 
trials. 

Questions. — Is  the  surface  tension  in  the  film  any  greater 
just  before  it  breaks  than  it  is  just  after  it  begins  to  form? 
Explain  your  answer. 

21.  BOYLE'S  LAW. 


To  show  that  at  a constant  temperature,  the  pressure 
of  a given  mass  of  gas  varies  inversely  as  its  volume.  (D. 
219,  221,  294,  295;  G.  168,  181,182;  K.  198-201;  W.  130,  133, 
134.) 

Apparatus. — Boyle's  law  apparatus,  barometer,  ther- 
mometer, pan. 


In  General  Physics 


117 


Theory,  Method  and  Description. — For  a given  mass 
of  gas  the  product  of  its  pressure  and  its  volume  varies  di- 
rectly as  its  absolute  temperature.  This  is  frequently  called 
the  gas  law.  If  p be  the  pressure,  v the  volume  and  T the 
absolute  temperature  of  a gas,  the  general  formula  for  the 
gas  law  may  be  written 

pv  — RT 

where  R is  a constant  for  the  given  gas.  Boyle ’s  law  is  a 
special  case  of  this  gas  law  and  holds  only  when  the  temper- 
ature remains  constant,  during  a change  of  pressure  or  of 
volume,  under  which  conditions  pv  equals  the  constant  RT. 
It  may  be  stated  as  follows: — The  temperature  of  a given  mass 
of  gas  remaining  constant , its  pressure  varies  inversely  as  its 
volume.  The  apparatus  for  obtaining  the  data  to  show  that 
this  law  holds  true  in  the  case  of  air,  consists  of  two  vertical 
glass  tubes  joined  at  the  bottom  by  a rubber  hose,  and  ad- 
justable in  height  on  vertical  supports.  This  apparatus  is 
similar  to  figure  32,  but  the  bulb  B is  replaced  by  a simple 
closed  tube,  the  other  tube  being  open.  The  tubes  are 
partly  filled  with  mercury,  confining  a quantity  of  air  in  the 
closed  tube,  and  the  height  of  the  mercury  columns  may  be 
read  on  a scale  between  the  supports  by  means  of  the  ad- 
justable indicator  and  vernier.  The  mercury  in  the  open 
tube  is  under  atmospheric  pressure  only,  but  by  varying  the 
height  of  the  two  tubes  the  pressure  on  the  confined  air  in 
the  closed  tube  may  be  varied.  By  varying  the  pressure 
and  volume  of  the  air  column  and  noting  the  heights  of  the 
mercury  columns,  and  the  corresponding  volume  of  the  air 
column,  as  well  as  the  barometric  pressure,  we  can  show  that 
the  pressure  times  the  volume  remains  constant,  provided  we 
take  care  to  keep  the  temperature  constant.  Instead  of 
finding  the  actual  volume  of  air  it  is  more  convenient  to 


118 


Manual  of  Experiments 


assume  the  cross  section*  of  the  tube  containing  the  air 
column  to  be  constant,  and  to  measure  the  length  of  this 
air  column  which  is,  of  course,  directly  proportional  to 
the  volume.  If  p,  v,  and  pr , v' , be  corresponding  values 
of  pressure  and  volume  for  two  different  lengths  l and  /' 
of  the  air  column  of  constant  cross  section  a , then  p v = pf  v' 
or  p a l = p'  a V from  which 

Pi  = P'  l' 

It  is  therefore  only  necessary  under  the  given  assumption 
to  measure  corresponding  values  of  pressure  and  length  of 
air  columns  to  verify  Boyle ’s  law.  When  the  mercury  col- 
umns are  of  equal  height,  the  pressure  on  the  air  columns 
is  that  of  the  atmosphere.  When  the  height  of  mercury 
in  the  open  tube  is  greater  than  that  in  the  closed  tube,  the 
pressure  on  the  confined  air  is  greater  than  one  atmosphere^ 
and  the  weight  of  a column  of  mercury  of  unit  cross  section 
and  having  a height  equal  to  the  difference  in  the  heights  of 
the  two  mercury  columns  must  be  added  to  the  atmospheric 
pressure  in  order  to  get  the  total  pressure  per  unit  area  on 
the  confined  air.  But  since  barometric  pressures  are  ex- 
pressed in  cm.  of  mercury,  we  need  only  add  the  difference 
in  height  of  the  two  mercury  columns  to  the  barometric 
height  to  express  the  pressure  on  the  air  enclosed  in  the  tube. 
If  the  height  of  mercury  is  greater  in  the  closed  tube,  the 
pressure  on  the  air  column  is  less  than  atmospheric  pressure 
because  the  latter  must  also  support  a column  of  mercury 
of  a height  equal  to  the  difference  of  the  two  mercury  levels. 
The  total  pressure  will  be  the  difference  between  the  bar- 
ometric height  and  the  difference  in  level  of  the  two  mercury 
columns. 

The  mercury  in  the  cistern  of  the  Fortin  barometer  must 
be  adjusted  by  means  of  the  screw  beneath  the  instrument 


d y z a 7~>  I 

L ' 

In  General  Physics  119 

until  the  surface  just  touches  the  ivory  tip  which  represents 
the  zero  of  the  scale.  The  barometer  height  should  be  care- 
fully read  on  the  attached  scale  and  vernier.  There  are  cer- 
tain corrections  to  be  made  in  this  reading.  Since  mercury 
in  a capillary  tube  is  depressed  below  its  free  level,  the  height 
of  mercury  in  the  barometer  tube  is  less  than  it  should  be 
by  an  amount  depending  on  the  diameter  of  the  tube.  This 
correction  should  be  added  to  the  barometer  reading.  Since 
on  the  other  hand  the  brass  scale  of  the  barometer,  and  the 
mercury  expand  with  an  increase  of  temperature,  the  density 
of  the  mercury  will  be  less,  and  the  reading  will  be  greater 
at  ordinary  temperatures  than  it  would  be  under  standard 
conditions;  that  is  at  0°  Centigrade.  We  must  therefore 
subtract  a certain  correction  from  the  barometric  height  as 
read  on  the  scale.  A table  of  these  corrections  will  be  found 
in  the  appendix. 

For  very  careful  work  a correction  should  also  be  made 
for  variations  in  the  acceleration  of  gravity  at  different  po- 
sitions on  the.  earth’s  surface,  but  the  correction  may  be 
neglected  in  the  present  case. 

In  this  particular  experiment  we  are  dealing  only  with 
relative  pressures,  and  as  the  barometer  is  at  practically  the 
same  temperature  as  the  other  mercury  columns  and  the 
expansion  of  the  brass  scale  is  small  compared  with  that  of 
the  mercury,  the  temperature  correction  may  be  omitted. 
The  two  glass  tubes  are  of  the  same  bore,  hence  the  cap- 
illary action  is  balanced  in  them  and  no  capillary  correction 
is  needed  in  the  heights  of  the  two  mercury  columns. 

Directions. — Caution. — Be  careful  not  to  lower  the 
open  tube  so  far  that  the  mercury  may  run  out.  Also  avoid 
handling  the  closed  tube  or  in  any  way  heating  it,  as  small 
changes  in  temperature  produce  an  appreciable  change  in 


120 


Manual  of  Experiments 


volume.  When  the  volume  of  the  air  column  has  been 
changed  by  altering  the  level  of  the  mercury,  always  give 
the  air  time  to  recover  from  the  resulting  change  in  temper- 
ature before  reading  the  height  of  the  mercury  columns. 

Attach  the  thermometer  to  the  closed  tube  in  order  to 
see  that  the  temperature  of  the  enclosed  air  remains  constant. 
Record  the  temperature  to  0.1°  and  see  that  it  is  the  same 
after  each  change  in  the  height  of  the  mercury  before  taking 
a reading.  Adjust  the  height  of  the  tubes  until  the  mercury 
stands  near  the  top  of  the  open  tube.  Read  to  0.01  cm.  the 
height  “a”  of  the  column  in  the  open  tube  and  the  height 
“b”  of  that  in  the  closed  tube.  Read  also  the  height  “c” 
of  the  top  of  the  enclosed  air  column.  Lower  the  mercury 
in  the  open  tube  about  10  cm.  and  read  the  new  heights  a 
and  b.  Continue  this  process  until  the  mercury  in  the  open 
tube  is  near  the  bottom.  Then  raise  the  mercury  again 
by  stages  of  about  10  cm.  each,  reading  the  corresponding 
heights  a and  b at  each  stage.  Read  the  barometer  height 
and  make  the  necessary  corrections  for  capillarity.  Cal- 
culate the  pressure  p = h + (a  — b)  at  each  position  (“ h ” 
being  the  corrected  barometric  height),  and  the  correspond- 
ing length  c — b.  With  this  data  show  that  Boyle’s  law 
holds  true. 

Question. — -a)  Show  the  form  of  the  graph  for  Boyle’s 
law  by  plotting  corresponding  values  of  pressure  and  volume 
of  your  data  on  coordinate  paper. 

b)  Plot  a curve  having  products  of  p and  v as  ordinates 
and  corresponding  values  of  p as  abscissae. 


In  General  Physics 


121 


Experiments  in  Heat. 

22.  CALIBRATION  OF  THERMOMETERS. 

(a)  To  determine  the  zero  and  steam  points  of  a ther- 
mometer. (b)  To  calibrate  the  scale  of  a thermometer. 
(D.  267;  G.  300-304;  K.  365-368;  W.  180.) 

Apparatus. — -A  100°C  thermometer,  a thermometer 
with  an  uncalibrated  scale,  hypsometer,  copper  calorimeter, 
ring  stand,  gas  tubing,  Bunsen  burner,  glass  funnel,  beaker. 

Description  and  Theory. — A mercury  thermometer 
consists  of  a thin  glass  bulb  connected  with  a capillary  tube. 
The  bulb  and  part  of  the  capillary  tube  are  filled  with  mer- 
cury and  then  the  tube  is  sealed  off  after  all  the  air  has  been 
expelled.  There  are  two  definite  temperatures  that  may  be 
used  to  locate  two  fixed  points  on  the  thermometer.  One 
is  that  of  melting  ice  which,  on  the  Centigrade  thermometer, 
is  marked  as  0°  and  the  other  is  the  temperature  of  vapor 
formed  over  water  boiling  freely  under  a pressure  of  76  cm. 
of  mercury.  This  point  is  marked  100°  on  the  Centigrade 
thermometer. 

When  an  ordinary  thermometer  is  used  for  any  sort  of 
accurate  work  it  is  generally  found  to  be  quite  badly  in  er- 
ror. Glass  is  very  unstable  in  its  molecular  structure.  As 
it  gets  older  the  molecules  gradually  readjust  themselves 
and  a sort  of  annealing  process  goes  on  similar  to  that  oc- 
curing  when  glass  is  gradually  cooled  after  being  worked  in 
a flame.  This  change  in  the  walls  of  the  thermometer  alters 
the  volume  contents  of  the  thermometer  bulb  and  tube,  so 
that  the  fixed  points  of  the  two  definite  temperatures  will 
be  found  to  have  changed.  In  making  standard  thermo- 


122 


Manual  of  Experiments 


meters  the  glass  must  be  cooled  very  slowly  and  allowed  to 
become  quite  old  before  they  are  calibrated,  in  order  that  the 
annealing  process  may  be  fully  completed.  Because  of  the 
change  in  the  boiling  and  freezing  points  of  the  thermo- 
meter, there  is,  of  course,  a corresponding  change  in  the 
position  of  the  other  divisions  of  the  thermometer. 

When  the  space  between  the  two  fixed  points  is  divided 
into  equal  divisions  the  bore  of  the  capillary  tube  is  sup- 
posed to  be  uniform  throughout.  This  is  not  really  true 
so  that,  for  very  accurate  work  it  is  necessary  to  calibrate 
and  make  a calibration  curve  for  the  thermometer. 

The  fixed  points,  the  0°  and  the  100°  points,  of  a thermo- 
meter are  tested  by  noting  the  readings  indicated  by  the 
thermometer  when  it  is  placed  in  melting  ice  and  dn  dry 
steam. 

If  t0  is  the  reading  of  the  calibrated  thermometer  when 
the  bulb  is  in  melting  ice  then  the  correction  at  zero  degrees  is 

C0  = [0°  - to]  . . . . (1) 

Again  if  the  temperature  of  steam  at  the  given  pressure  is 
Ta  and  the  reading  of  the  thermometer  when  placed  in  the 
steam  is  ta,  the  correction  at  the  boiling  point  is 

“ [Tg : ~ ta]  . . . . (2) 

Assuming  the  bore  of  the  calibrated  thermometer  to  be 
uniform,  which  is  allowable  for  ordinary  work,  the  correction 
for  any  intermediate  temperature  tt  is  given  approximately 
by  x 


Ct  = [C0  + 


cs-c 


o 


100 


Therefore  the  correct  temperature  tc,  when  the  thermometer 
reads  tt,  is  given  by 


4 = 4 + + 


Cs-C0 


4] 


100 


(4) 


In  General  Physics 


123 


The  zero  point  and  the  steam  point  of  the  uncalibrated 
thermometer  can  be  located  in  the  same  manner  as  the  corres- 
ponding points  of  the  calibrated  thermometer.  The  inter- 
mediate points  can  be  calibrated  by  placing  the  two  ther- 
mometers in  the  same  vessel  of  water  at  various  tempera- 
tures and,  having  observed  the  readings  of  both,  by  de- 
signating the  point  on  the  uncalibrated  thermometer  by  the 
corrected  reading  of  the  calibrated  thermometer  for  each 
temperature  of  the  water  in  which  they  are  immersed. 

The  hypsometer,  Fig.  31b,  used  in  locating  the  steam 


point,  is  a copper  vessel  having  a reservoir  in  the  lower  part 
for  holding  water  and  in  the  upper  part  having  an  inner  and 


124 


Manual  of  Experiments 

an  outer  jacket.  The  two  jackets  are  so  constructed  that  the 
steam  generated  in  the  reservoir  has  to  pass  up  through  the 
inner  jacket  and  down  through  the  space  between  the  two 
jackets  before  escaping  through  the  outlet  tube.  If  the 
thermometer  is  inserted  in  a cork  in  the  top  of  the  hypso- 
meter  it  is  thus  surrounded  by  dry  steam. 

Directions. — The  freezing  point  should  be  tested  first. 
To  do  this,  arrange  a glass  funnel  in  a ring  support  as  shown 
in  Fig.  31a,  with  a beaker  beneath  to  catch  the  melted  ice. 
Fill  the  funnel  with  snow  or  cracked  ice  and  insert  the  ther- 
mometer so  that  the  entire  thread  of  mercury  is  beneath  the 
surface  of  the  melting  mixture.  After  the  thermometer  has 
been  in  the  mixture  long  enough  for  its  reading  to  reach  the 
melting  temperature,  take  readings  tQ,  every  minute  until 
three  readings  have  been  obtained.  Read  carefully  to  tenths 
of  a degree.  At  the  time  of  each  reading  raise  the  ther- 
mometer so  that  the  top  of  the  thread  of  mercury  may  be 
seen  just  above  the  surface  of  the  cracked  ice  and  then  quick- 
ly take  an  accurate  reading. 

In  order  to  test  the  boiling  point,  pour  water  into  the 
reservoir  of  the  hypsometer  till  it  is  about  two-thirds  full 
and  place  a Bunsen  flame  under  it.  This  may  be  done  while 
the  freezing  point  is  being  tested.  Do  not  let  the  hypsometer 
boil  dry.  Insert  the  thermometer  in  the  hypsometer  as 
shown  in  Fig.  31b,  so  that  the  100°  point  may  be  seen  just 
above  the  top  of  the  cork.  After  the  steam  has  been  escap- 
ing freely  for  some  time  from  the  tube  leading  from  the  outer 
jacket,  and  the  mercury  in  the  thermometer  has  reached 
a steady  point,  take  readings  ts  every  minute  until  three 
readings  have  been  obtained.  Read  carefully  to  tenths  of 
a degree. 

Read  the  barometer  and,  having  corrected  the  reading 


In  General  Physics 


125 


according  to  the  table  of  corrections  in  the  Appendix,  find 
the  boiling  temperature  Ta,  corresponding  to  that  corrected 
pressure,  by  referring  to  the  steam  table  in  the  Appendix. 

It  is  found  that  after  the  thermometer  has  been  ex- 
posed to  the  steam  the  glass  bulb  does  not  contract  to  its 
former  volume  for  some  time  afterwards.  So  that,  if  the 
thermometer  is  placed  again  in  the  melting  ice  (after  being 
allowed  to  cool  30°  or  40°  below  the  boiling  point)  the  po- 
sition of  its  freezing  point  will  have  changed  slightly. 

After  the  thermometer  has  acquired  the  temperature 
of  the  freezing  mixture  take  readings  every  minute  until 
three  readings  have  been  obtained.  In  the  calibration  of 
the  uncalibrated  thermometer  the  average  of  these  readings 
will  be  taken  as  the  freezing  point  tQ  of  the  calibrated  ther- 
mometer instead  of  that  obtained  before  the  thermometer 
was  heated.  Substitute  in  equations  (1)  and  (2)  and  find 
the  corrections  CQ  and  Cs. 

In  a similar  manner  note  the  readings  indicated  by  the 
uncalibrated  thermometer  when  placed  first  in  dry  steam 
and  then  in  melting  ice.  Draw  a scale  in  your  note  book 
similar  to  the  one  upon  which  the  thermometer  is  mounted 
and  locate  upon  the  scale  the  position  of  the  temperature  of 
steam  and  the  zero  point  just  observed. 

In  order  to  fix  intermediate  points  on  the  uncalibrated 
thermometer,  place  the  two  thermometers  in  a calorimeter 
containing  water  at  about  20°C.  Observe  the  position  of 
the  mercury  column  when  they  have  become  stationary, 
and  having  corrected  the  reading  tt  of  the  calibrated  ther- 
mometer according  to  equation  (4),  record  this  corrected 
reading  on  the  scale  in  your  note  book  at  the  position  indi- 
cated by  the  mercury  column  on  the  uncalibrated  thermo- 
meter scale.  In  a similar  manner  locate  and  determine  the 
value  of  the  position  on  the  uncalibrated  scale  when  the  filler- 


126 


Manual  of  Experiments 


mometers  are  immersed  in  water  at  temperatures  of  about 
40°,  60°  and  80°C. 

Questions. — What  is  the  difference  between  the  aver- 
ages of  the  two  sets  of  readings  obtained  for  the  freezing 
point  before  and  after  the  boiling  point  readings?  Explain 
the  cause  of  this  difference. 


23.  CHARLES J LAW. 

Pressure  Coefficient  of  Air. 

To  show  that  the  volume  of  a given  mass  of  gas  remain- 
ing constant,  the  pressure  varies  directly  as  its  absolute  tem- 
perature. (D.  264,  265,  279,  293,  294;  G.  332,  333;  K.  376, 
393-397;  W.  195,  197.) 

Apparatus. — Air  thermometer,  hypsometer,  tripod,  Bun- 
sen burner,  copper  can,  stirrer,  mercury  thermometer,  and 
cloth. 

Theory,  Method  and  Description. — (See  theory  under 
experiment  al.)  Charles’  law  (also  called  Gay  Lussac’s 
law)  is  a special  case  of  the  gas  law,  and  holds  only  when 
the  pressure  of  a gas  remains  constant  while  its  volume  and 
temperature  vary.  Under  these  conditions  the  ratio  of  vol- 
ume to  absolute  temperature  is 

v/T  = R/p  (a  constant). 

% 

The  law  may  therefore  be  stated  as  follows: — The  pressure 
of  a given  mass  of  gas  remaining  constant , its  volume  is  di- 
rectly proportional  to  its  absolute  temperature.  It  is  quite 
evident  that,  when  the  volume  of  the  gas  is  kept  constant, 


In  General  Physics 


127 


its  pressure  is  directly  propor+ional  to  its  absolute  temper- 
ature, or 

p /T  = R /v  { a constant). 

This  may  be  looked  upon  as  another  way  of  expressing 
Charles  ’ law,  and  it  is  this  law  which  we  wish  to  verify. 
The  gas  constant  R may  be  shown  to  be  equal  to  a p0v0 
where  pc  and  vQ  are  the  pressure  and  volume  respectively  of 
the  given  mass  ,of  gas  at  zero  degrees  centigrade  and  a is 
the  pressure  coefficient , or  that  fractional  part  of  the  pressure 
at  zero  degrees  Centigrade  by  which  it  increases  for  each  degree 
Centigrade  rise  in  temperature . That  is  to  say,  the  pressure 
of  the  given  mass  of  gas  at  t degrees  Centigrade  will  be 

Pt  ~Po  (1  + at) 

and 

a = (pt  — po)  / (po  t) 

or  the  pressure  coefficient  will  be  the  change  in  pressure  per 
unit  pressure  at  zero  degrees  Centigrade  per  degree  Centi- 
grade change  in  temperature. 

From  the  relation  pt  = pQ  (1  + at)  we  see  that  at  the 
temperature 

t = — 1/a, 

pt  will  equal  zero.  This  point  in  the  scale  of  temperatures ’ 
at  which  there  is  no  energy  in  the  gas , is  called  the  absolute 
zero)  and  temperatures  measured  from  this  point  are  called 
absolute  temperatures.  On  the  Centigrade  scale  the  ab- 
solute zero  is — 273°,  and 

T°  (abs,)  = (*+273)°C.  * 

The  apparatus  used  to  verify  this  law  is  similar  to  the 
Boyle ’s  law  apparatus,  but  the  closed  tube  leads  into  a hori- 
zontal capillary  tube  a foot  or  more  in  length  which  ends  in 


128 


Manual  of  Experiments 


a vertical  elongated  bulb,  B.  This  arrangement  makes  it 
possible  to  place  the  bulb  in  baths  at  different  temperatures, 
and  also  keeps  these  baths  at  some  distance  from  the  mer- 
cury columns.  See  Fig.  32.  The  volume  of  air  can  be 
kept  constant  by  adjusting  the  mercury  each  time  after  the 
air  has  come  to  the  temperature  of  the  bath,  so  that  it  stands 
at  a mark  on  the  apparatus  just  at  the  beginning  of  the  cap- 
illary tube.  The  temperature  of  the  air  in  this  tube  will  be 


\ 


i 


Fig.  32 


slightly  different  from  that  of  the  bath,  but  its  volume  is 
so  small  in  comparison  with  the  volume  of  air  in  the  bulb, 


In  General  Physics 


129 


that  we  may  consider  all  of  the  air  to  be  at  the  same  temper- 
ature. The  volume  of  the  bulb  does  not  remain  absolutely 
constant  but  increases  as  the  glass  expands  with  a rise  in 
temperature,  and  as  the  pressure  of  the  confined  air  increases. 

By  observing  the  temperatures  of  the  baths,  and  the  cor- 
responding pressures  due  to  the  atmospheric  pressure  on  the 
open  tube  and  the  difference  in  level  of  the  mercury  columns, 
we  obtain  the  data  for  verifying  the  law  and  calculating  the 
pressure  coefficient. 

The  law  holds  only  approximately  for  gases  which  do 
not  obey  Boyle ’s  law,  but  for  air  the  change  in  the  ratio  p/T 
is  very  slight  and  the  pressures  may  be  used  to  find  the  cor- 
responding temperatures  over  a wide  range  by  means  of  a 
calibration  curve.  The  apparatus  is  therefore  often  called 
an  air  thermometer. 

Directions. — Adjust  the  mercury  columns  until  the 
height  of  that  in  the  open  tube  is  at  least  15  cm.  less  than 
that  in  the  closed  tube.  Place  the  copper  can  (without 
stirrer)  on  a stand  so  that  the  bulb  projects  into  it,  being  care- 
ful not  to  break  the  latter  nor  the  capillary  tube.  Fill  the 
can  with  snow  or  finely  crushed  ice  until  the  bulb  is  covered. 
Fill  the  interstices  of  the  ice  with  cold  water.  Allow 
several  minutes  for  the  air  in  the  bulb  to  come  to  the  tem- 
perature of  the  ice  bath,  i.  e.  0°  C.  Adjust  the  tubes  until 
the  mercury  is  at  some  well  defined  mark  near  the  begin- 
ning of  the  capillary  tube,  being  careful  that  it  does  not 
run  into  the  capillary  tube  and  the  bulb.  Note  the  height 
of  the  mercury  column  11  a”  in  the  open  tube  and  the  height 
ilb”  in  the  closed  tube  and  record  the  difference  in  level 
(a  — b).  Remove  the  ice  from  the  can  and  replace  it  in  the 
ice  box  at  the  sink.  Fill  the  can  with  water  and  heat  it  to 
about  25  degrees  C.  Then  remove  the  burner  and  allow 
the  water  and  the  air  in  the  bulb  to  come  to  a common  tern- 


130 


Manual  of  Experiments 


perature,  using  the  stirrer  to  equalize  the  temperature  of  the 
water.  Note  this  temperature.  Adjust  the  height  of  the 
mercury  so  that  it  stands  at  the  same  mark  on  the  closed 
tube  as  before,  in  order  to  make  the  volume  of  air  the  same, 
and  observe  the  new  difference  in  level  of  the  mercury  col- 
umns. Then  heat  the  water  to  about  60  degrees  and  pro- 
ceed to  find  the  nev  difference  in  level  at  this  temperature. 
Then  carefully  lower  the  level  of  the  mercury  column  until 
its  height  in  the  open  tube  is  again  the  first  value  of  “a”, 
before  removing  the  hot  bath,  so  that  when  the  air  in  the 
bulb  cools  and  contracts,  the  mercury  will  not  run  over  into 
the  bulb.  Next  fill  the  can  of  the  hypsometer  about  one- 
half  full  of  water,  place  it  on  the  tripod  so  that  the  bulb  of 
thermometer  projects  into  the  hypsometer,  and  adjust  a cloth 
around  the  neck  of  the  bulb  to  prevent  steam  from  escaping 
freely.  Then  heat  the  hypsometer  until  steam  is  generated 
freely  and. again  adjust  the  mercury  columns  until  the  vol- 
ume of  air  is  the  same  as  it  was  before.  Observe  the  differ- 
ence in  level  of  the  mercury  columns  and  also  the  corrected 
barometric  pressure  h.  (See  experiment  21  for  corrections 
to  barometric  readings.)  Before  lowering  the  temperature 
of  the  enclosed  air  by  removing  the  steam  bath,  be  sure  to 
lower  the  level  of  the  mercury  to  the  original  value  “a”,  so 
that  no  mercury  flows  into  the  bulb  when  the  air  contracts. 

Obtain  the  temperature  of  the  steam  at  the  barometric 
pressure  from  the  steam  table  in  the  appendix,  and  calculate 
the  total  pressure  corresponding  to  each  temperature  used. 

With  the  data  taken  show  that  Charles 1 law  holds  true, 
and  calculate  the  pressure  coefficient  of  air  for  constant  vol- 
ume, and  the  absolute  zero. 

Comparison  of  Temperatures  by  the  Air  and  the  Mercury 
Thermometers. — With  the  data  taken  for  melting  ice  and  for 
steam,  plot  a straight  line  curve  having  total  pressure  of 


In  General  Physics 


131 


enclosed  air  as  ordinates  and  corresponding  temperatures 
as  abscissae,  assuming  the  pressure  coefficient  to  be  a con- 
stant throughout  this  range  of  temperatures.  On  this  curve 
find  the  temperatures  corresponding  to  the  pressures  ob- 
served when  the  bulb  was  immersed  in  the  baths  at  about 
25  and  60  degrees  C.  respectively.  Compare  each  of  these 
temperatures  with  the  corresponding  reading  on  the  mercury 
thermometer.  Find  also  the  absolute  zero  on  the  Centigrade 
scale  by  prolonging  the  curve  until  it  meets  the  axis  of  tem- 
peratures. 

Question. — What  are  the  reasons  for  and  against  the 
use  of  the  air  thermometer  as  compared  with  the  mercury  in 
glass  thermometer? 

24.  LINEAR  EXPANSION. 

To  determine  the  coefficient  of  linear  expansion  of  a met- 
al rod.  (D.  273-275;  G.  314-321;  K.  379-388;  W.  184-186.) 

Apparatus. — Linear  expansion  apparatus  and  metal  rod, 
steam  generator,  Bunsen  burner,  tripod,  thermometer,  mi- 
crometer screw,  funnel  and  waste  water  jar. 

Theory,  Method  and  Description. — The  coefficient 
of  linear  expansion  of  a substance  is  the  change  in  length  of 
unit  length  of  the  substance  for  a change  in  temperature  of  one 
degree  Centigrade . For  instance  if  a metal  rod  changes 
from  a length  Lx  cm.  to  L2  cm.  when  the  temperature  changes 
from  h to  t2,  the  total  change  in  length  per  cm.  of  its  original 
length  is  (L2  — L{)  /Li  and  the  change  per  degree  change  in 
temperature  will  be 

_ L2—Li 
a ” Uih-h). 

For  isotropic  bodies  this  coefficient  is  the  same  in  all  di- 
rections. Strictly  speaking  it  is  not  a constant  but  depends 


132 


Manual  of  Experiments 


upon  the  temperature,  and  in  order  to  define  it  precisely  we 
should  refer  to  the  change  in  length  of  unit  length  of  the 
substance  at  zero  degrees  Centigrade.  For  all  practical 
purposes  however,  (since  the  change  with  temperature  is 
very  small)  a is  calculated  by  the  formula  given  above. 

The  method  used  is  to  place  a metal  rod  about  one  meter 
long  in  a bath  of  cold  water,  noting  the  temperature  of  the 
water  and  measuring  the  length  of  the  rod  when  it  has  come 
to  the  same  temperature;  then  to  replace  the  water  bath  by 
one  of  steam,  again  noting  the  temperature  and  meas- 
uring the  length  of  the  rod  after  allowing  sufficient  time  for 
it  to  reach  the  temperature  of  the  steam.  With  this  data 
the  coefficient  of  linear  expansion  may  be  calculated  by  the 
formula  given. 

The  metal  rod  is  surrounded  by  a sheet  copper  jacket, 
its  ends  projecting  through  rubber  corks  which  fit  tightly 
about  the  rod  and  within  the  ends  of  the  jacket.  See  Fig.  33. 


The  jacket  is  covered  with  sheet  asbestos  to  prevent  loss  of 
heat  by  radiation,*  etc.,  and  rests  upon  a wooden  support. 
At  one  end  of  this  wooden  support  is  an  adjustable  screw  S 
which  may  be  turned  into  contact  with  the  rod.  At  the 


In  General  Physics 


133 


other  end  is  a micrometer  screw  M which  can  also  be  turned 
into  contact  with  the  rod  and  serves  to  measure  its  elong- 
ation. To  measure  the  temperature  of  the  bath  and  rod 
a thermometer  T is  passed  through  a cork  in  an  opening 
at  the  middle  of  the  upper  side  of  the  jacket.  A second 
opening  at  one  end  serves  to  introduce  the  water  and  steam 
and  a third  opening  at  the  other  end  of  the  under  side  is 
fitted  with  a rubber  tube  for  carrying  off  the  water  or  steam 
and  may  be  closed  by  means  of  a pinch  clamp. 

Directions. — To  save  time  the  water  may  be  heated 
to  the  boiling  point  while  the  rest  of  the  experiment  is  being 
performed.  Fill  the  steam  generator  about  one-half  full 
of  water,  and  heat  it  with  a Bunsen  burner,  placing  it  far 
enough  from  the  rod  to  prevent  heating  the  latter.  Mean- 
time close  the  lower  outlet  with  the  pinch  clamp  and  fill  the 
jacket  with  cold  water,  being  careful  not  to  cause  an  over- 
flow. Adjust  the  thermometer  in  the  upper  opening  so  that 
it  rests  on  the  rod  and  allow  sufficient  time  for  the  rod  and 
the  water  to  come  to  a common  temperature  before  noting 
the  latter.  Adjust  the  brass  set  screw  so  that  it  touches 
the  end  of  the  rod,  then  adjust  the  micrometer  screw  until 
it  is  about  five  millimeters  from  the  end  of  the  rod  and 
the  zero  of  the  graduated  circular  disc  is  at  the  edge  of  the 
straight  mm.  scale.  Then  turn  the  screw  into  contact  with 
the  rod,  counting  the  number  of  turns  and  noting  the  number 
of  divisions  on  the  last  fraction  of  a turn.  (Since  there  are 
100  divisions  on  the  circular  scale  9 complete  turns  and  45 
divisions  over  would  give  945  divisions,  etc.)  In  order  to 
tell  when  the  screw  and  rod  come  in  contact  an  electric 
buzzer  is  used.  (See  experiment  13.)  Turn  the  micro- 
meter back  to  the  starting  point  but  leave  the  set  screw  as 
it  is.  Measure  the  length  of  the  bar  in  cm.  with  the  beam 
compass.’  Allow  the  water  to  run  out  into  the  sink  and 


134 


Manual  of  Experiments 


leave  the  lower  outlet  open.  When  steam  is  being  freely 
generated,  connect  the  generator  to  the  upper  end-opening 
by  means  of  rubber  tubing  and  allow  it  to  flow  through  the 
jacket,  noting  the  change  of  temperature  on  the  thermometer. 
When  steam  issues  freely  from  the  lower  outlet  and  the 
temperature  remains  constant  at  about  98°,  see  that  the  set 
screw  is  in  contact  with  the  rod.  If  it  is  not  in  contact, 
do  not  move  the  screw  but  press  the  rod  gently  into  contact. 
Then  turn  the  micrometer  screw  until  it  just  touches  the 
end  of  the  rod,  noting  the  number  of  divisions  through  which 
you  turn  it.  Then  record  the  temperature  of  the  bar  in 
the  steam.  The  pitch  of  the  micrometer  screw  is  1/2  mm. 
hence  there  are  2000  of  the  divisions  on  the  circular  scale  in 
a cm.  Find  the  expansion  of  the  rod,  the  difference  of  the 
micrometer  readings  in  cold  water  and  in  steam.  From  the 
data  obtained  calculate  the  coefficient  of  expansion  of  the 
rod. 

Repeat  the  experiment,  and  calculate  the  mean  coefficient 
from  the  two  determinations. 

Question. — a)  How  would  your  result  be  affected  if 
the  rod  were  not  in  contact  with  the  screw  S after  expansion. 

b)  What  force  in  pounds  weight  would  be  needed  to 
stretch  the  rod  the  same  amount  as  in  this  experiment? 

25.  COEFFICIENT  OF  CUBICAL  EXPANSION  OF 

A LIQUID. 

To  determine  the  coefficient  of  cubical  expansion  of  a li- 
quid. (D.  277;  G.  327,  328;  K.  389;  W.  189-191.) 

Apparatus. — Specific  gravity  bottle,  liquid,  thermometer, 
fine  balance,  set  of  known  masses,  calorimeter,  tripod, 
Bunsen  burner. 


In  General  Physics 


135 


Description  and  Theory. — The  specific  gravity  bottle 
(Fig.  34.)  is  simply  a small  bottle  which  is  provided  with  an 
accurately  fitting  ground  glass  stopper.  A very  small  hole 
through  the  center  of  this  stopper  leads  to  the  interior  of  the 
bottle,  its  object  being  to  allow  the  bottle  to  be  completely 
filled  with  any  liquid. 

The  method  used  in  this  experiment  to  find  the  coefficient 
of  cubical  expansion  of  a liquid,  depends  upon  the  relative 
expansion  of  the  liquid  and  glass.  Let  m0  be  the  mass  of  the 


bottle  and  liquid  at  0°C,  d0  the  density  of  the  liquid  at  the 
same  temperature  and  m the  mass  of  the  bottle.  Then  the 
volume  of  the  liquid  at  0°C  will  be 


Mo  — m 


Vo  = 


a) 


If  j3  is  the  coefficient  of  cubical  expansion  of  the  liquid,  its 
volume  at  t°C  will  be 

Mo  — M 


V = Vo(l  +(lt)  = 


(1+180 


(2) 


136 


Manual  of  Experiments 


Similarly,  the  capacity  of  the  bottle  at  0°C  is 

Mo  — m 


Vo 


. (1) 


and,  if  the  coefficient  of  cubical  expansion  of  glass  is  a , its 
capacity  at  f C will  be 


Mo  — M 

vf  — Vo  (1  + at)  — (1  at) 

do 


(3) 


The  volume,  at  f C,  of  the  liquid  expelled  is  therefore 

Mo  ~ M Mo  — M Mo—M 

■(1+00 U—  (1  + at)  = ; 1 (fi-a).  (4) 


dc. 


do 


'YYLq  — 

This  volume  is  also  given  by  — — — (1  where  Mi  is  the 

dQ 

mass  of  the  bottle  and  liquid  at  the  temperature  t.  There- 
fore, 


Mo  — Ml  Mo  — M 

— — — (1  +00  = ■ ■ t (13  — a). 

do  do 

Solving  (5)  for  j3,  we  find 


(5) 


Mo  — Mi  Mo  — M 

0=7 H oi  (6) 

(Mi  — M)t  Mi~M 

Since  m0  — m and  mi  — m differ  very  little,  equation  (6) 
may  be  written. 

Mo  ~ Mi 

(3  = — — + a . . (7) 

(Mi~M)t 

This  equation  expresses  the  coefficient  of  cubical  ex- 
pansion of  the  liquid  in  terms  of  the  coefficient  of  cubical 
expansion  of  glass,  the  original  mass  of  liquid  in  the  bottle 
at  zero  degrees,  the  final  temperature  \ f,  and  the  mass  of  the. 
liquid  that  is  expelled  from  the  bottle  when  its  temperature 
is  changed  from  0°  to  t° C. 


In  General  Physics 


137 


Directions. — Clean  and  dry  the  specific  gravity  bottle 
very  carefully,  then  find  its  mass  m on  the  fine  balance.  Fill 
the  bottle  with  the  liquid  whose  coefficient  of  expansion  is 
to  be  determined,  place  it  in  the  calorimeter  and  pack  snow 
or  shaved  ice  around  it.  See  that  none  of  the  ice  gets  into 
the  bottle.  Place  the  stopper  also  in  the  ice,  and  after 
leaving  the  bottle  and  stopper  in  the  ice  for  10  minutes,  dry 
the  stopper  quickly,  but  thoroughly  with  a cloth  and  put 
it  in  the  bottle.  See  that  no  air  bubbles  cling  to  the  stopper. 
Leave  a small  drop  of  the  liquid  on  the  top  of  the  stopper  so 
as  to  have  the  capillary  tube  filled  in  case  there  is  a further 
contraction  of  the  volume  of  the  liquid  in  the  bottle.  Leave 
the  bottle  in  the  ice  at  least  5 minutes  longer,  then  wipe 
off  the  top  of  the  stopper  with  a cloth.  Remove  the  bottle, 
dry  it  carefully,  and  then  find  the  mass  mQ  of  the  bottle 
and  contents  on  the  fine  balance.  The  weighing  should  be 
done  as  quickly  as  possible,  as  a precaution  against  the  loss 
of  weight  through  evaporation  of  the  liquid  that  may  escape 
as  its  temperature  begins  to  rise. 

Heat  some  water  (not  closer  than  to  within  10  or  12 
degrees  of  the  boiling  point  of  the  liquid  whose  coefficient  is 
being  determined,)  and  place  the  bottle  in  it.  Do  not  let 
the  water  extend  above  the  neck  of  the  bottle.  After  leaving 
the  bottle  in  the  water  10  or  15  minutes,  note  the  temperature 
t of  the  water,  wipe  off  the  liquid  from  the  stopper,  remove 
the  bottle,  and  having  dried  it  thoroughly,  find  its  mass  mi 
on  the  balance.  Taking  the  value  of  a the  coefficient  of 
cubical  expansion  of  glass,  as  0,000025,  substitute  the  values 
found  for  m , mQ , mi  and  t , in  equation  (7)  and  calculate  the 
value  of  p. 

In  a similar  manner  make  a second  determination  of  the 
value  of  p.  * 

Questions. — If  a bubble  of  air  is  trapped  in  the  bottle 


138 


Manual  of  Experiments 


when  making  the  weighing  mQ,  what  effect  will  it  have  on 
the  value  of  (3  found?  Explain  your  answer. 

CALORIMETRY. 

* 

(D.  280-284;  G.  452-458;  K.  398-401;  W.  199-201.) 

In  each  of  the  following  four  experiments  a quantity  of 
heat  is  to  be  measured.  The  calorimeters  in  which  this 
measurement  is  to  be  made,  are  metal  vessels,  usually  cov- 
ered with  insulating  material  to  prevent  the  loss  of  heat  by 
radiation,  and  supplied  with  a stirrer  for  equalizing  the  tem- 
perature of  the  water  which  they  contain. 

The  method  usually  employed  is  known  as  “the  method 
of  mixtures,  ” which  is  based  upon  the  principle  of  equal 
heat  exchanges.  That  is,  when  two  or  more  bodies,  orig- 
inally at  different  temperatures,  are  placed  in  thermal  con- 
tact, and  the  heat  exchanges  take  place  exclusively  between 
these  bodies,  the  quantity  of  heat  lost  by  one  part  of  the  sys- 
tem of  bodies  is  equal  to  the  heat  gained  by  the  other  part. 
In  this  method  a given  mass  M grams  of  the  substance  whose 
heat  is  to  be  determined  is  heated  to  a temperature  t2° C. 
and  is  quickly  plunged  into,  or  mixed  with  W grams  of  water 
at  a lower  temperature  t° C.  The  mixture  will  then  come 
to  a uniform  temperature  f C. 

It  is  evident  that  besides  the  gain  in  heat  by  the  water 
in  rising  to  the  temperature  of  the  mixture,  the  calorimeter 
containing  the  water  and  any  other  bodies  such  as  the  ther- 
mometer and  stirrer,  which  have  a common  temperature 
with  the  water  will  also  gain  in  heat.  The  simplest  way  of 
taking  these  bodies  into  account  is  to  treat  them  as  so  much 
extra  water  beyond  the  amount  W)  that  is,  we  must  find 
their  water  equivalent.  The  water  equivalent  of  a body  is 
the  amount  of  water  which  requires  as  much  heat  to  raise 


In  General  Physics 


139 


its  temperature  i°C.  as  does  the  body  in  question.  It  is  equal 
to  the  mass  of  the  body  multiplied  by  its  specific  heat. 

In  order  to  avoid  errors  due  to  gain  or  loss  of  heat  by 
radiation  from  or  to . neighboring  bodies,  the  following 
method  of  compensation  is  generally  used.  On  starting, 
the  water  and  calorimeter  are  brought  to  a certain  tempera- 
ture below  that  of  the  room  and  the  process  of  heating  is 
continued  until  they  are  at  a temperature  as  much  above 
that  of  the  room,  as  they  were  originally  below  it.  In  this 
way  approximately  as  much  heat  is  gained  by  the  calorimeter 
from  the  surrounding  air  until  it  comes  to  the  same  tem- 
perature as  the  latter,  as  is  lost  by  it  to  the  air,  while  it 
exceeds  this  temperature. 

26.  SPECIFIC  HEAT. 

To  determine  the  specific  heat  of  copper  by  the  method 
of  mixtures.  (D.  280-284;  G.  452-454,  457-466;  K.  398-406; 
W.  199-207.) 

Apparatus. — Calorimeter,  steam  generator,  steam  heater, 
two  thermometers,  copper  or  aluminum  pellets,  drying  cup, 
tripod,  wooden  paddle,  paper  funnel,  cloth,  coarse  balance 
and  set  of  weights. 

Theory,  Method  and  Description. — Every  substance 
requires  a definite  amount  of  heat  to  raise  a given  mass  of 
the  substance  through  a given  temperature.  The  quantity 
of  heat  necessary  to  raise  the  temperature  of  one  gram  of  the 
substance  through  one  degree  C.,  at  any  given  temperature , 
is  called  the  specific  heat  of  the  substance  at  that  tempera- 
ture. For  scientific  purposes  the  quantity  is  measured  in 
terms  of  a unit  called  the  calorie.  The  calorie  may  be  de- 
fined as  the  quantity  of  heat  required  to  raise  the  tempera- 
ture of  one  gram  of  water  from  15°  to  16°  C.  (This  is  the 


140 


Manual  of  Experiments 

mean  heat  capacity  of  water  between  its  freezing  and  boiling 
points).  The  specific  heat  of  water  at  15°  C.  is  therefore 
unity. 

In  engineering  practice  quantity  of  heat  is  measured  in 
British  thermal  units.  The  B.  T.  U.  is  the  heat  required  to 
raise  the  temperature  of  one  pound  of  water , one  degree 
Fahrenheit. 

The  specific  heat  of  copper  or  of  aluminum  is  to  be  de- 
termined by  the  method  of  mixtures.  Let  W be  the  known 
mass  of  water  in  the  calorimeter,  at  a temperature  h,  M the 
mass  of  the  substance  whose  specific  heat  C is  to  be  deter- 
mined, h its  temperature,  t the  temperature  of  the  mixture, 
and  w the  water  equivalent  of  the  calorimeter. 

Then  taking  the  specific  heat  of  water  as  unity,  the  heat 
gained  will  be 

(W-\-w)  (t  — ti)  calories. 

In  falling  to  the  temperature  of  the  mixture,  the  heated 
body  will  lose  an  amount  of  heat 

C M (t2  — t)  calories. 

Equating  the  amounts  of  heat  lost  and  gained,  we  have 
C M (fk  — t)  = (W+w)  ( t-h ) 

from  which  to  calculate  the  specific  heat. 

The  water  equivalent  of  the  thermometer  is  small  and 
may  be  neglected,  and  if  the  calorimeter  cup  and  stirrer  are 
of  the  same  material  as  the  pellets,  we  may  assume  that  they 
have  the  same  specific  heat.  If  their  joint  mass  is  m,  their 
water  equivalent  will  be  Cm  and  the  equation  becomes 

C M (t2-t)  = ( W+Cm ) (t-h) 

from  which  we  calculate  C 

To  prevent  heat  becoming  lost  by  radiation  the  copper 
calorimeter  cup  is  surrounded  by  a hair-felt  jacket  enclosed 
within  a second  copper  vessel.  The  calorimeter  has  a wood- 


In  General  Physics 


141 


en  lid  which  holds  a copper  stirrer  for  equalizing  the  tem- 
perature of  the  water.  A thermometer  passes  through  a 
rubber  stopper  in  the  middle  of  this  lid  and  through  the 
stirrer.  The  metal  pellets  are  heated  in  a cup  which  is 
surrounded  by  a steam  jacket. 

Directions. — Fill  the  steam  generator  about  half  full 
of  water  and  heat  it  with  a Bunsen  burner.  Take  enough 
metal  pellets  to  fill  the  inner  calorimeter  cup  about  one- 
third  full,  and  pour  them  into  the  small  cup  of  the  steam 
heater.  The  pellets  should  be  perfectly  dry.  # If  they  are 
not,  pour  them  into  the  brass  cup,  and  heat  them  over  the 
flame,  stirring  meanwhile  with  the  wooden  paddle  until  they 
are  dry.  When  they  have  cooled  somewhat  pour  them  into 
the  small  steam  heater  cup  and  insert  the  cork  and  thermo- 
meter. Work  the  bulb  of  the  thermometer  into  the  pellets, 
being  careful  not  to  break  it.  Connect  tU^ojjrer  opening 
of  the  steam  heater  to  the  steam  generator^tu  allow  steam 
to  come  out  through  the  opposite  upperflTole,  so  that  it  passes 
all  around  the  small  can.  Close  the  outer  lower  opening 
with  a rubber  hose  and  pinch  clamp.  After  the  steam  be- 
gins to  issue  shake  the  heater  occasionally  and  invert  it,  so 
that  the  pellets  are  thoroughly  mixed  and  the  temperature 
uniform  throughout,  being  careful  to  hold  the  cork  and  ther- 
mometer in  place.  After  a time  the  temperature  will  be- 
come stationary  at  97°  or  98°C.  In  the  meantime  weigh 
the  inner  calorimeter  cup  (dry)  and  record  the  mass  in  grams. 
Fill  the  cup  a little  more  than  half  full  of  ice  water  and  weigh 
again.  Place  cork  and  thermometer  in  place  and  adjust 
the  length  of  the  thermometer  so  that  it  will  reach  within 
about  two  centimeters  of  the  bottom  of  the  cup.  Place  the 
cup  in  the  calorimeter  and  see  that  there  is  no  ice  in  the  water. 
Shake  the  steam  heater  once  more  and  read  the  temperat  ure 
to  one-tenth  degree.  Then  remove  the  cork  and  with  the 


142 


Manual  of  Experiments 


help  of  the  paper  funnel  pour  the  pellets  into  the  ice  water 
with  one  swoop.  Do  not  delay  to  collect  stray  pellets  but 
immediately  insert  the  cork  and  thermometer,  being 
careful  not  to  break  the  latter  by  forcing  it  into  the 
pellets.  Shake  the  calorimeter  for  a few  seconds,  being  care- 
ful not  to  spill  any  of  the  water,  and  watch  the  thermometer. 
Just  as  it  ceases  rising  and  begins  to  drop  read  the  temper- 
ature to  one  tenth  degree.  Weigh  the  inner  calorimeter 
cup  again,  with  water,  and  pellets,  included.  From  the  data 
taken  calculate  the  specific  heat  of  the  metal  pellets. 

Before  repeating  the  determination  note  whether  the 
temperature  of  the  mixture  was  as  much  above  that  of  the 
room  as  the  original  temperature"" was  below  it,  and  if  not, 
determine  by  an  inspection  of  the  data  whether  more  cold 
water  is  needed  or  more  hot  pellets,  to  obtain  this  temper- 
ature. With  the  altered  amount  of  water  or  pellets,  repeat 
the  experiment  and  calculate  again  the  specific  heat  of  metal. 

Questions. — a)  What  is  the  effect  on  your  result,  of 
spilling  some  of  the  water  from  the  calorimeter  while  shaking 
it?  . 

b)  Which  metals  heat  up  the  quicker,  those  with  high 
or  those  with  low  specific  heat? 

27.  THE  MECHANICAL  EQUIVALENT  OF  HEAT. 

To  determine  the  mechanical  equivalent  of  heat  by  means 
of  Callendar’s  apparatus.  (D.  280-282,  290;  G.  452-454, 
505,  506;  K.  409-411;  W.  199-201,  249-251,  494.) 

Apparatus. — Calendar’s  apparatus  with  two  2000  gram 
weights,  coarse  balance  and  set  of  weights,  beaker,  funnel, 
bent  thermometer,  straight  thermometer,  six  50  gram  weights 
and  hose  and  nozzle  for  draining. 


In  General  Physics 


143 


Theory,  Method  and  Description. — The  first  law  of 
thermodynamics  may  be  stated  as  follows: — Whenever  me- 
chanical energy  is  converted  into  heat , or  heat  into  mechanical 
energy , the  ratio  of  the  mechanical  energy  to  the  heat  is  constant. 
This  constant  ratio  is  called  the  mechanical  equivalent  of 
heat.  It  is  the  number  of  units  of  work  which  must  be  done 
in  order  to  produce  unit  quantity  of  heat  energy.  To  de- 
termine this  constant  we  shall  use  the  following  meth- 
od:— A cylindrical  brass  calorimeter  containing  a known 
quantity  “ W ” of  water  at  a known  temperature  (Cti” 
is  rotated  against  the  friction  of  silk  bands  which  bear 
upon  it  owing  to  known  weights  which  are  suspended  from 
them.  On  starting,  the  water  and  calorimeter  are  at 
a certain  temperature  below  that  of  the  room,  and  the  cal- 
orimeter is  turned  until  it  is  at  a temperature  as  much  above 
that  of  the  room  as  it  was  below  at  the  start.  The  total 
number  of  revolutions  tcn”  of  the  calorimeter  and  the  new 
temperature  li  W’  of  the  water  are  noted  and  the  circum- 
ference “ cv  of  the  calorimeter  measured.  Since  the  calori- 
meter also  absorbs  heat  in  rising  to  this  temperature  we  must 
calculate  its  water  equivalent  lizv”  or  the  amount  of  water 
which  would  require  just  as  much  heat  as  the  calorimeter 
does  to  be  raised  through  unit  temperature.  If  the  aver- 
age force  at  the  circumference  of  the  calorimeter  is  the  weight 
of  a mass  m,  the  work  done  in  overcoming  this  resistance 
through  a distance  equal  to  n times  the  circumference  c is 
n c m g.  The  quantity  of  heat  required  to  raise  the  water 
and  calorimeter  from  the  temperature  h to  the  temperature 
h will  be  {W -\-w)  ( h~h ).  The  ratio  of  these  two  quanti- 
ties will  be  the  mechanical  equivalent  of  heat  “J,”  or 

J (W -\-w)  (h  — tf)  = n c m g. 

The  apparatus  (See  Fig.  35)  consists  of  the  mounted 
cylindrical  brass  calorimeter  C rotated  by  hand  by  means  of 


144 


Manual  of  Experiments 


a crank  wheel,  the  number  of  revolutions  being  registered 
by  a counter  on  one  side.  Unequal  weights  Mi , M2  are  sus- 
pended from  the  ends  of  a silk  belt  which  bears  on  the  cyliji- 


Fig.  35 


der  and  makes  one  and  a half  complete  turns  around  it.  A 
light  spring  balance  B is  arranged  so  that  it  acts  in  direct 
opposition  to  the  lighter  weight.  The  motion  of  the  belt  is 
limited  by  means  of  stops,  and  the  weights  are  adjusted  and 
the  cylinder  rotated  so  that  the  weights  are  held  suspended, 


In  General  Physics 


145 


the  final  adjustment  being  made  automatically  by  the  spring 
balance.  The  temperature  of  the  water  is  read  on  a bent 
thermometer  T inserted  through  a central  opening  on  the 
front  end  of  the  cylinder. 

Directions. — Hang  the  4000  grams  from  the  end  of 
the  belt  opposite  the  spring  balance  and  add  small  brass 
weights  to  the  other  end  until  “floating  equilibrium M is 


maintained  when  the  cylinder  rotates;  that  is  until  the 
weights  are  not  supported  at  the  stops,  but  hang  freely  from 
the  belt.  Level  the  apparatus  so  that  no  part  of  this 


146 


Manual  of  Experiments 


floating  system  touches  the  frame.  The  student  chosen  to 
rotate  the  cylinder  should  start  it  a number  of  times  to  be- 
come familiar  with  its  behavior.  Hang  the  straight  ther- 
mometer near  the  calorimeter,  allow  it  to  come  to  room 
temperature  and  note  this  temperature.  Place  about  200 
grams  of  distilled  water  in  a beaker  and  cool  it  to  about  7° 
C.  below  the  room  temperature.  Weigh  funnel,  beaker 
and  water,  then  pour  the  water  into  the  calorimeter  through 
the  central  opening,  and  reweigh  funnel  and  beaker  with 
whatever  water  may  be  adhering  to  them.  Pass  the  bent 
thermometer  through  the  central  opening  (being  careful 
not  to  put  any  strain  on  it  by  pressing  against  the  walls  of 
the  cylinder, )and  fasten  it  so  that  its  bulb  is  immersed  in 
the  water.  Remove  the  weights  from  the  belt  and  turn  the 
cylinder  until  the  thermometer  indicates  a steady  tem- 
perature, after  which  replace  the  weights.  Then  rotate 
the  cylinder  with  a steady  motion.  A second  student  should 
note  the  number  of  revolutions  on  the  counter  and  at  every 
45th  and  95th  revolution  he  should  call  “ ready  ”,  and  at 
every  50th  and  100th  revolution  he  should  call  “read”. 
A third  student  should  note  the  temperature  of  the  water 
at  the  word  “read”  and  a fourth  student  should  record 
all  data.  The  cylinder  should  be  rotated  until  the  tem- 
perature of  the  water  is  about  7°  C.  above  the  tempera- 
ture of  the  room.  The  water  should  then  be  drawn  from  the 
calorimeter  by  removing  the  flat  headed  brass  screw  and  re- 
placing it  by  the  nozzle  and  hose.  Calculate  the  length  of 
the  circumference  of  the  cylinder  by  winding  a string  sev- 
eral times  around  it,  measuring  its  length,  and  finding  the 
length  of  a single  turn.  The  average  force  exerted  at  the 
circumference  c will  be  the  weight  of  the  4000  grams  M\g 
plus  the  spring  balance  reading  Msg  minus  the  small  brass 
weights  M2£.  (See  Fig.  36.) 


In  General  Physics 


147 


With  the  data  obtained  calculate  the  mechanical  equiv- 
alent of  heat,  assuming  g = 980  cm/sec.2. 

Question. — If  it  requires  4.2  joules  to  raise  one  gram 
of  water  1°C.  in  temperature, what  is  the  value  of  “J”  a), 
in  foot  pounds  per  calorie?  b).  In  foot  pounds  per  B.T.U? 
c)  In  foot  poundals  per  B.  T.  U.? 


28.  LATENT  HEAT  OF  FUSION. 


To  determine  the  latent  heat  of  fusion  of  ice.  (D.  280- 
283,  307,  G.  452-458,  467,  K.  398-401,  424-426,  W.  199-201, 
208-211.) 

Apparatus. — Calorimeter  with  lid  and  stirrer,  hot  water 
reservoir  with  lid  and  stirrer,  water  heater,  thermometer, 
ring  stand,  tripod,  Bunsen  burner,  coarse  balance,  set  of 
weights  and  drying  cloths. 

Theory,  Method  and  Description. — If  a solid  be  heat- 
ed slowly,  when  its  melting  point  is  reached  it  will  be  found 
that  the  temperature  remains  constant  at  this  point  until 
all  the  substance  is  changed  into  the  liquid  state,  all  of  the 
heat  supplied  being  used  up  in  this  change  of  state.  Each 
substance  requires  a definite  amount  of  heat  to  transform  a 
given  amount  of  the  substance  from  the  solid  to  the  liquid 
state  at  the  temperature  of  the  melting  point.  The  latent 
heat  of  fusion  of  the  substance  is  the  number  of  calories  of  heat 
required  to  change  one  gram  of  the  substance  at  the  melting 
point  from  the  solid  to  the  liquid  state. 

To  determine  the  latent  heat  of  fusion  of  ice  we  may  use 
the  method  of  mixtures  (see  Calorimetry.)  A mass  M 
grams  of  ice  at  0°  C.  is  added  to  a mass  W grams  of  water  at 
h°C.  in  a copper  calorimeter.  The  temperature  t of  the 
water  must  be  noted  at  the  instant  when  all  of  the  ice  is 
melted.  The  specific  heat  of  copper  may  be  taken  as 


148  Manual  of  Experiments 

0.093  calories  per  gram  per  degree  Centigrade,  and  if  the 
mass  of  the  calorimeter  and  stirrer  is  m grams,  their  water 
equivalent  w will  be 

w = 0.093  m grams 

Since  the  specific  heat  of  water  is  unity  the  quantity  of  heat 
gained  by  the  ice  in  melting  and  rising  to  the  temperature 
of  the  mixture  after  it  is  melted,  will  be  M — ) 

calories,  where  L{  is  the  latent  heat  of  fusion  of  the  ice.  The 
heat  lost  by  the  water  and  calorimeter  in  dropping  to  the 
temperature  t will  be  (W-\-w)  ( fa  — t ) calories.  Equating 
these  two  quantities  we  have 

M ( Lf+t ) = (W+w)  (fe-0 
from  which  we  calculate  Lf. 

To  avoid  error  in  the  result,  the  ice  should  be  carefully 
dried  before  placing  it  in  the  calorimeter.  To  avoid  error 
due  to  radiation,  the  final  temperature  of  the  water  should 
be  as  much  below,  as  its  original  temperature  was  above 
that  of  the  room. 

The  calorimeter  consists  of  a copper  cup  with  a wooden 
lid  and  copper  stirrer,  surrounded  by  a hair  felt  jacket  en- 
closed within  a second  copper  vessel. 

The  hot  water  reservoir  is  a copper  vessel  with  lid  and 
stirrer,  covered  with  a hair  felt  jacket  and  having  at  the 
bottom  an  outlet  for  the  water.  The  lids  and  stirrers  of 
the  calorimeter  and  reservoir  are  pierced  with  holes  to  ac- 
commodate the  thermometers. 

Directions. — Fill  the  steam  heater  about  three  fourths 
full  of  hot  water  and  heat  to  about  60°.  Run  this  hot  water 
into  the  reservoir,  being  careful  to  have  the  lower  opening 
closed  with  the  rubber  tubing  and  pinch  clamp.  Weigh  out 
about  60  to  80  grams  of  ice  in  lumps  about  the  size  of  wal- 
nuts, and  place  them  on  a strip  of  cloth.  Note  the  temper- 
ature of  the  room.  Dry  the  inner  calorimeter  cup  and  stirrer 


In  General  Physics 


149 


with  the  hot  air  blast  and  weigh  them.  Then  run  hot  water 
into  the  cup  and  add  cold  water  until  the  temperature  is 
about  fifteen  degrees  above  that  of  the  room.  Have  the 
water  fill  the  cup  about  four-fifths  full.  While  one  student 
weighs  the  calorimeter  with  water  and  stirrer,  as  quickly  as 
possible,  the  other  should  dry  the  lumps  of  ice  with  the  strip 
of  cloth.  After  stirring  a few  seconds  to  equalize  the  tem- 
perature of  the  water,  read  the  thermometer  to  0.1°  and 
quickly  insert  the  lumps  of  ice  under  the  stirrer.  Stir  the 
water  and  note  the  change  of  temperature  as  the  ice  melts. 
If  the  ice  is  all  melted  before  the  temperature  is  as  much 
below,  as  it  originally  was  above  that  of  the  room,  or  if  there 
are  still  lumps  of  ice  present  when  this  temperature  is  reached, 
treat  this  as  a preliminary  trial  from  which  to  determine  the 
proper  amount  of  ice  to  use.  With  this  amount  of  ice  pro- 
ceed as  before  and  when  the  ice  is  just  melted  read  the  ther- 
mometer quickly.  Weigh  the  calorimeter  with  the  stirrer, 
water  and  melted  ice  and  from  the  data  taken  calculate  the 
latent  heat  of  fusion  of  ice.  Repeat  the  experiment,  using 
more  or  less  ice,  or  colder  or  hotter  water  to  fulfill  the  given 
conditions  if  necessary. 

Question. — How  would  your  results  be  affected  a)  if 
the  ice  were  not  all  melted  when  the  temperature  of  the 
mixture  was  taken?  b)  If  an  appreciable  amount  of  the 
ice  had  melted  between  weighing  it  and  placing  it  in  the 
calorimeter?  Give  reasons  for  your  answers. 

29.  LATENT  HEAT  OF  VAPORIZATION. 

To  determine  the  latent  heat  of  vaporization  of  water. 
(D.  280-283,  308,  314;  G.  452-454,  457,  458,  468;  K.  398- 
401,  432-442;  W.  199-201,  213,  214.) 

Apparatus. — Calorimeter  with  stirrer  and  glass  tube, 


150 


Manual  of  Experiments 


water  trap,  connecting  hose,  steam  boiler,  thermometer,  bar- 
ometer, tripod,  Bunsen  burner,  brass  cup,  coarse  balance, 
set  of  weights,  and  cloth. 

Theory,  Method  and  Description. — -If  a liquid  be 
heated  slowly,  when  its  boiling  point  is  reached  it  will  be 
found  that  the  temperature  remains  constant  at  this  point 
until  all  the  liquid  has  been  changed  to  a vapor,  all  of  the 
heat  supplied  being  used  up  in  this  change  of  state.  Each 
substance  requires  a definite  amount  of  heat  to  transform  a 
given  amount  of  the  substance  from  the  liquid  to  the  vapor 
state  at  the  temperature  of  the  boiling  point.  The  latent 
heat  of  vaporization  of  the  substance  is  the  number  of  calories 
of  heat  required  to  change  one  gram  of  the  substance  at  the 
boiling  point  from  the  liquid  to  the  vapor  state . 

To  determine  the  latent  heat  of  vaporization  of  water, 
we  may  use  the  method  of  mixtures.  Steam  is  passed  into 
W grams  of  water  at  a temperature  t\  degrees  (below  that 
of  the  room)  in  a calorimeter  until  the  temperature  t of  the 
water  is  as  much  above,  as  it  originally  was  below  that  of  the 
room.  The  mass  M of  the  condensed  steam  is  obtained  from 
the  increase  in  mass  of  the  water.  The  calorimeter  consists 
of  a copper  cup  surrounded  by  a hair  felt  jacket  and  enclosed 
within  a second  copper  vessel.  It  has  a wooden  lid  and  a 
copper  stirrer  for  equalizing  the  temperature  of  the  water 
which  it  contains.  If  the  mass  of  the  calorimeter  cup  and 
stirrer  is  m and  their  specific  heat  0.093  calories  per  gram 
per  degree  Centigrade,  their  water  equivalent  w will  be 

w = 0.093  m grams. 

The  temperature  h of  the  steam  may  be  obtained  from 
the  barometric  pressure  at  the  time  of  the  experiment 
by  referring  to  the  steam  table  in  the  Appendix.  Since 
the  specific  heat  of  water  is  unity,  the  heat  gained  by  the 
calorimeter  and  water  will  be  {W  + w)  {t—tf)  calories, 


In  General  Physics 


151 


and  the  heat  lost  by  the  steam  in  condensing  and  by  the  re- 
sulting water  in  dropping  to  the  temperature  of  the  mixture 
will  be  M Ly  + M (h  — t)  calories,  where  Lw  is  the  latent 
heat  of  vaporization.  Equating  these  two  quantities  we  get 

M (Xv-|-^2  — t)  = (JV -{-zv)  (t  — 1\) 

from  which  we  calculate  LY. 

The  temperatures  t and  h are  chosen  equidistant  above 
and  below  the  room  temperature  respectively  in  order  to 
avoid  errors  due  to  radiation. 

Any  drops  of  water  which  may  be  carried  into  the  cal- 
orimeter by  the  steam  will  of  course  produce  an  error  in  the 
result,  for  the  heat  of  vaporization  of  these  drops  has  not 
been  available  for  raising  the  temperature  of  the  water  in  the 
calorimeter.  To  avoid  this  error  the  steam  is  passed  through 
a water-trap  before  entering  the  calorimeter.  This  trap 
consists  of  a glass  tube  about  2.5  cm,  in  diameter  and  10  cm. 
in  length,  closed  at  the  ends  by  corks.  The  steam  enters 
and  leaves  the  trap  by  glass  tubes  which  pass  eccentrically 
through  the  corks.  They  are  placed  out  of  line  so  that  the 
steam  can  not  pass  directly  from  one  into  the  other  with 
its  full  velocity,  carrying  moisture  with  it.  The  outlet  tube 
extends  nearly  to  the  top  of  the  trap,  the  free  end  passing 
vertically  through  the  lid  of  the  calorimeter  to  within  about 
rfone  centimeter  of  the  bottom.  Any  moisture  carried  over 
by  the  steam  or  condensing  in  the  tube  will  collect  in  the 
lower  part  of  the  trap. 

Directions. — Fill  the  steam  generator  about  half  full 
of  water  and  heat  it.  Dry  the  inner  calorimeter  cup  and 
stirrer  and  weigh  them  on  the  coarse  balance.  Note  the 
temperature  of  the  room  and  the  barometric  pressure,  on 
the  thermometer  and  barometer  in  the  laboratory.  Fill  the 
calorimeter  about  four-fifths  full  of  water  and  cool  it  with 


152 


Manual  of  Experiments 


lumps  of  ice  to  about  10°  below  the  room  temperature.  Then 
remove  any  ice  which  may  be  left  in  the  calorimeter,  insert 
the  stirrer,  and  carefully  weigh  the  cup  with  water  and  stirrer. 
When  steam  issues  freely  from  the  generator,  connect  the 
latter  to  the  steam  trap  by  means  of  the  rubber  tube,  and 
allow  the  steam  to  flow  for  a few  minutes.  Note  the  tem- 
perature of  the  water  and  calorimeter,  remove  any  moisture 
that  may  have  collected  in  the  trap,  by  taking  out  one  of 
the  corks,  replace  the  cork  and  quickly  insert  the  tube 
through  the  lid  and  stirrer.  Stir  the  water  while  the  steam 
is  condensing  and  note  the  rise  in  temperature.  When  the 
water  is  about  10°  above  room  temperature,  quickly  remove 
the  tube,  stir  the  water  and  note  the  temperature  when  it 
reaches  its  maximum.  Then  weigh  the  cup  with  stirrer, 
water,  and  condensed  steam.  From  the  data  taken  cal- 
culate the  latent  heat  of  vaporization  of  water. 

Repeat  the  experiment  four  or  five  times,  varying  the 
amounts  of  the  different  factors  as  may  be  found  necessary, 
and  using  freshly  cooled  water  each  time. 

Question. — How  would  your  results  be  affected  a)  if 
an  appreciable  amount  of  solder  (specific  heat  = 0.04)  forms 
part  of  the  weight  of  the  calorimeter  cup?  b)  If  the  steam 
were  introduced  into  the  calorimeter  at  a depth  of  15  cm. 
under  the  surface  of  the  water?  Give  reasons  for  your 
answers. 

30.  HYGROMETRY.— HUMIDITY  AND  DEW  POINT. 

To  determine  the  dew  point  and  relative  humidity  of 
the  air  in  the  laboratory.  (D.  309;  G.  437-442;  K.  447,  448; 
W.  220,  221; 

Apparatus. — Nickel  plated  cup,  copper  stirrer,  vessel 


In  General  Physics 


153 


for  holding  ice  water,  two  thermometers,  cloth,  wet  and  dry 
bulb  hygrometer  (mounted  in  room.) 

Description  and  Theory. — The  atmosphere  consists 
of  air  mixed  with  water  vapor.  Ordinarily  it  is  not  satur- 
ated though  saturation  may  be  brought  about  either  by  an 
increase  in  the  amount  (mass  per  unit  volume)  of  water 
vapor  without  change  of  temperature,  or  from  a lowering 
of  the  temperature  without  change  in  the  amount  of  vapor 
present.  An  hygrometer  is  an  instrument  designed  for  the 
determination  of  the  amount  of  water  vapor  in  the  air.  Two 
of  the  most  common  types  are  the  dew  point  hygrometer 
and  the  wet  and  dry  bulb  hygrometer.  The  dew  point  is 
the  temperature  to  which  the  atmosphere  must  be  cooled 
in  order  that  it  may  be  saturated  by  the  amount  of  water 
vapor  actually  present.  In  its  simplest  form  the  dew  point 
hygrometer  consists  of  a metal  vessel  whose  outer  surface 
is  polished.  By  adding  a cold  mixture  to  the  vessel,  its  tem- 
perature may  be  reduced  until  moisture  from  the  atmosphere 
is  deposited  on  the  polished  surface.  The  temperature,  2d,  at 
which  this  occurs  is  the  dew  point.  Knowing  this  and  the 
temperature  of  the  air,  the  absolute  and  relative  humidities 
may  be  obtained  from  hygrometry  tables.  The  absolute 
humidity  is  the  number  of  grams  of  water  contained  in  one 
cubic  meter  of  air.  The  relative  humidity  is  the  ratio  of  the 
mass  of  water  vapor  actually  present  in  a cubic  meter  of  air 
to  the  mass  which  a cubic  meter  of  air  would  contain  if  satu- 
rated at  the  temperature. 

The  wet  and  dry  bulb  hygrometer  consists  of  two  sensitive 
thermometers  mounted  on  the  same  stand,  the  bulb  of  one 
being  covered  with  cloth  which  is  kept  continually  moist 
by  being  connected  with  a vessel  of  water.  Unless  the  at- 
mosphere is  saturated  with  moisture  there  will  be  constant 


154 


Manual  of  Experiments 


evaporation  from  the  cloth  covering  the  bulb  and  this  will 
cause  the  wet  bulb  thermometer  to  indicate  a lower  tem- 
perature than  the  other.  The  drier  the  air  the  more  rapidly 
will  evaporation  take  place  and  the  greater  will  be  the  dif- 
ference between  the  readings  of  the  two  thermometers. 
With  these  readings  the  various  hygrometric  values  such  as 
dew  point,  relative  humidity,  etc.,  may  be  obtained  from 
tables  constructed  from  the  results  of  experiment. 

Directions. — -Note  the  temperature  of  the  laboratory 
near  the  place  where  the  experiment  is  to  be  performed. 
Make  a mixture  of  crushed  ice  and  water  in  the  cup  for  ice 
water.  Clean  and  polish  the  nickel  cup,  pour  water  from 
the  faucet  into  it  to  a depth  of  about  a centimeter  and  then 
add  ice  water  slowly  until  dew  forms  on  the  polished  surface. 
In  the  meantime,  stir  the  mixture  with  the  stirrer.  Do  not 
breathe  on  the  polished  surface  as  this  will  cause  a film  of 
moisture  to  appear.  In  case  the  depth  of  the  water  in  the 
cup  gets  to  be  more  than  about  three  centimeters  without 
dew  forming,  pour  out  part  of  the  water  and  continue  to  add 
ice  water  as  before.  When  dew  finally  forms  it  will  cover 
the  polished  surface  to  the  height  that  the  water  stands  on 
the  inside.  Note  the  temperature  of  the  water  at  the  first 
appearance  of  the  dew.  Now  let  the  water  warm  up  or,  if 
need  be,  add  faucet  water  in  small  portions  to  the  mixture, 
noting  the  temperature  when  the  dew  begins  to  disappear. 
The  mean  of  these  two  readings  will  give  the  temperature  of 
the  dew  point. 

Repeat  the  above  determination  two  more  times,  drying 
and  polishing  the  can  each  time.  Take  the  average  of  the 
three  values  for  the  temperature  of  the  dew  point. 

In  case  the  dew  point  is  below  0°C,  the  mixture  will 
have  to  be  cooled  by  the  addition  of  salt  and  shaved  ice. 


In  General  Physics 


155 


Refer  to  the  hygrometry  table  in  the  appendix,  and 
determine  the  saturation  humidity  at  the  temperature 
of  the  dew  point.  This  is  evidently  the  absolute  humidity 
of  the  air  in  the  room.  Calculate  the  relative  humidity. 

Now  take  the  thermometer  readings  on  one  of  the  wet 
and  dry  bulb  hygrometers  in  the  room  (the  one  nearest  the 
place  where  you  have  determined  the  dew  point)  after  first 
being  sure  that  the  cloth  is  wet.  In  order  to  get  the  proper 
depression  the  wet  bulb  thermometer  should  be  fanned  or 
some  other  device  should  be  employed  for  maintaining  a 
rapid  motion  of  the  air  past  the  covered  bulb.  If  this  is 
not  done  the  air  in  the  immediate  neighborhood  of  the  ther- 
mometers will  soon  become  more  moist  than  the  bulk  of  the 
air  in  the  room  and  evaporation  will  take  place  less  rapidly 
than  it  should. 

Record  the  reading  of  the  wet  bulb  thermometer  as  tW} 
that  of  the  dry  bulb  thermometer  as  t,  and  the  reading  of  the 
barometer  as  b.  From  the  hygrometry  table  find  pw,  the 
maximum  vapor  pressure  for  water  vapor  at  the  temperature 
tw.  The  actual  pressure  p exerted  by  the  water  vapor  present 
in  the  atmosphere  (at  temperature  t)  depends  upon  the 
density  of  the  air  into  which  evaporation  takes  place,  that 
is,  upon  the  barometric  height  b}  as  well  as  upon  the  velocity 
of  the  air  currents.  The  empirical  formula 

p — pw  — 0.00075  b {t  — tw) 

gives  the  pressure  approximately  when  the  air  surrounding 
the  thermometers  is  in  moderate  motion.  In  the  above 
formula  p,  pw  and  b are  expressed  in  cm.  of  mercury  and 
(t  — *w)  in  degrees  centigrade. 

Having  found  p,  the  dew  point  td  (the  temperature  at 
which  this  pressure  produces  saturation)  may  be  obtained 


t 


156 


Manual  of  Experiments 


from  the  hygrometry  table.  Compute  the  relative  humidity 
of  the  air  in  the  room  from  the  data  obtained. 

Questions. — (1)  Explain  why  drops  form  on  the  out- 
side of  a pitcher  containing  cold  water.  Does  the  saying 
that  this  11  indicates  an  approaching  rainstorm  ” have  any 
scientific  foundation?  Explain  your  answer.  (2)  What  will 
be  the  effect  on  the  final  result  for  the  relative  humidity  if 
the  breath  of  the  observer  strikes  the  wet  bulb  thermometer 
when  the  data  is  being  taken? 


In  General  Physics 


157 


Experiments  in  Magnetism  and  Electricity. 


To  study  the  field  of  force  about  magnets  and  about  con- 
ductors carrying  a current  of  electricity.  (D.  367,  368,  426, 
427;  G.  699,  700,  833,  836;  K.  486,  487,  674,  676,  679;  W. 
418-420,  472,  476,  513,  514.) 

Apparatus. — U shaped  magnet,  bar  magnet,  six  pieces 
of  cardboard,  five  paper  outlines  of  size  and  shape  of  magnets, 
six  sheets  of  blue  print  paper  in  a closed  tube,  six  clips, 
pins,  iron  filings  in  a sifter,  rectangular  and  circular  coils 
of  wire,  solenoid,  reversing  key  and  small  compass,  con- 
necting leads  with  plug  and  lamp  resistance. 

Theory  and  Method. — A magnetic  field  of  force  is 
the  space  surrounding  a magnet  or  a conductor  carrying  a 
current  of  electricity,  throughout  which  it  exerts  a magnetic 
force.  We  may  think  of  this  space  as  threaded  with  lines  of 
magnetic  force.  The  latter  may  be  defined  as  lines  drawn  in 
the  field,  such  that  at  every  point  of  a line,  the  direction  of  the 
resultant  magnetic  force  at  that  point  is  tangent  to  the  line. 
The  direction  of  such  a line  is  the  direction  in  which  a free 
north-seeking  magnetic  pole  would  move.  Hence  the  lines 
pass  from  the  N pole  of  a magnet  to  the  S pole  through  the 
surrounding  medium.  When  iron  filings  are  brought  into  this 
field,  each  small  piece  of  iron  becomes  a temporary  magnet 
by  induction,  and  places  itself  tangent  to  a line  of  force 
with  its  north-seeking  pole  pointing  in  the  direction  of  the 
line.  The  filings  may  be  sprinkled  upon  sensitive  paper 
in  the  field  and  exposed  to  sunlight.  If  the  filings  are 
then  removed  and  the  paper  developed,  the  image  formed 


31.  MAGNETIC 


158 


Manual  of  Experiments 


by  the  shadow  of  the  filings  will  show  the  shape  and  density 
of  the  lines  of  force. 

The  lines  of  force  about  a straight  conductor  carrying 
a current  of  electricity  are  concentric  circles.  If  the  con- 
ductor be  grasped  in  the  right  hand  with  the  thumb  point- 
ing in  the  direction  of  the  current,  the  fingers  will  encircle 
the  conductor  in  the  direction  of  the  lines  of  force.  If  the 
conductor  is  wound  in  the  form  of  a flat  coil  the  lines  of  force 
about  it  are  no  longer  concentric  circles,  but  a large  number 
of  them  will  encircle  all  of  the  loops.  If  the  conductor  is 
wound  in  a solenoid  (a  long  cylindrical  coil  with  an  axis 
much  greater  than  its  diameter)  the  field  of  force  is  still 
more  distorted  and,  on  the  outside,  resembles  closely  that  of 
a bar  magnet,  the  lines  passing  from  one  face  (or  end)  to 
the  other  face  of  the  coil  and  back  to  the  first  face  through 
the  coil. 

Directions. — Caution. — Do  not  expose  the  blue  print 
paper  to  the  light  any  longer  than  is  necessary  but  keep  it 
in  the  closed  tube,  taking  out  one  sheet  at  a time.  To  get 
the  proper  time  of  exposure  for  this  paper,  cut  a strip  about 
1/4  inch  wide  from  one  of  the  sheets  and  expose  a piece  an 
inch  or  two  long  to  bright  light  (sunlight  if  possible)  until 
it  assumes  a bronzed  appearance.  Then  wash  for  a few 
minutes  in  clean  water.  If  the  print  is  a dark  blue  it  has 
been  exposed  sufficiently  long;  if  light  blue  the  exposure  has 
been  too  short;  if  grey  blue  it  has  been  too  long.  In  either 
of  the  last  two  cases  make  a second  or  third  trial  and  note 
carefully  the  time  of  exposure  required  to  obtain  a dark 
blue  print. 

Fasten  a sheet  of  blue  print  paper  flat  upon  a cardboard 
by  means  of  pins  at  the  corners  and  place  it  symmetrically 
over  the  bar  magnet  in  a place  shaded  from  the  sunlight. 


In  General  Physics 


159 


(See  Fig.  37).  Sprinkle  iron  filings  over  the  paper  from  the 
sifter,  holding  the  latter  about  a foot  above  the  paper  in 
order  to  distribute  them  more  uniformly.  At  the  same  time 
tap  the  cardboard  gently  with  a pencil  so  that  the  filings 
arrange  themselves  along  the  lines  of  force,  until  the  latter 
are  distinctly  outlined.  If  the  filings  are  sprinkled  on  too 
thickly  the  lines  will  not  be  distinct.  Next  pin  to  the  board 


just  over  the  magnet  a piece  of  thin  card  cut  to  the  same  size 
as  the  magnet,  with  a + and  a—  cut  out  at  the  ends,  the 
+ end  coinciding  with  the  N pole  or  marked  end  of  the  mag- 
net and  the  — end  with  the  S pole.  Lift  up  the  cardboard 
in  a vertical  direction  from  the  magnet  and  place  it  carefully 
in  the  bright"  light  at  the  window,  exposing  it  for  the  proper 
length  of  time  to  get  a dark  blue  print.  Then  pour  the 
filings  carefully  into  the  pan  provided  at  the  sink,  remove 
the  paper  from  the  cardboard,  write  your  name  upon  the 
back  in  lead  pencil,  and  place  it  face  down  in  the  water  for 
about  ten  minutes  or  until  the  unchanged  sensitizer  has  been 
washed  out.  In  the  meantime  repeat  the  performance,  using 
the  bar  magnet  placed  about  4 cm.  in  front  of  the  U shaped 
magnet  with  like  poles  opposite  each  other.  Then  reverse 
the  bar  magnet  and  repeat  again.  Finally  make  a duplicate 


160 


Manual  of  Experiments 


set  of  these  three  prints  so  that  each  student  has  a set  to 
bind  in  with  his  written  report  of  the  experiment. 

When  the  prints  are  thoroughly  washed  out,  hang  them 
up  to  dry.  Before  binding  them  in  your  report  mark  arrow 
heads  on  ten  or  twelve  distinct  lines  of  force  on  each  print, 
showing  their  direction. 

Join  the  loose  ends  of  the  lamp  cord  to  the  terminals 
at  one  end  of  the  reversing  switch  and  place  the  porcelain 
plug  in  the  Chapman  receptacle  in  (or  by)  the  bench.  Always 
join  conductors  to  terminals  marked  with  the  same  sign 
(+  or  — ).  The  current  will  flow  from  the  + to  the  — ter- 
minal. The  reversing  switch  has  its  end  terminals  cross 
connected  in  such  a way  that  when  the  source  of  current  is 
connected  to  the  middle  terminals  and  the  apparatus  through 


(a)  (b) 

Fig.  38a  and  b 


which  the  current  is  to  be  sent  is  connected  to  the  two  ter- 
minals at  either  end,  or  vice  versa,  the  current  through  the 
apparatus  will  be  in  one  direction  when  the  switch  is  closed 
through  one  end,  and  reversed  when  the  switch  is  closed 
through  the  other  end  terminals.  Join  the  middle  terminals 
of  the  switch  to  the  rectangular  coil  as  shown  in  the  diagram, 
Fig.  38a.  A lamp  joined  in  this  circuit  serves  to  cut  down 
the  current  which  would  otherwise  flow  through  the  coil  and 
prevents  short  circuiting  when  the  loose  terminals  are  acci- 


In  General  Physics 


161 


dently  brought  into  contact.  Set  the  rectangular  coil  with 
its  plane  in  the  north  and  south  meridian,  and  close  the 
switch.  Place  the  small  compass  on  the  shelf  at  the  north' 
side  of  the  coil  and  move  it  around  the  conductor,  noting 
on  the  data  sheet  the  directions  of  the  lines  of  force  on  each 
side  by  means  of  small  arrows.  The  dots  at  b,  d,  etc.,  on  the 
data  sheet  represent  cross  sections  of  the  wire  conductors. 
Do  the  same  for  the  conductor  at  the  south  end  of  the  coil. 
Then  place  the  compass  above  and  below  the  horizontal  por- 
tions of  the  coil  at  a and  at  c and  note  the  direction  of  the 
lines  of  force.  Next  reverse  the  direction  of  the  current  in  the 
coil,  by  closing  the  switch  in  the  opposite  direction,  and  re- 
peat the  observation. 

Replace  the  rectangular  coil  by  the  two  circular  coils 
(Fig.  38b.)  and  connect  the  coils  so  that  the  current  of  elec- 
tricity flows  through  both  in  the  same  direction.  With  the 
aid  of  the  compass  determine  the  direction  of  the  lines  of  force 
at  the  sides  where  the  coils  pierce  the  supporting  table,  and 
note  the  direction  of  the  current  in  “the  coils  at  these  points 
(whether  up  or  down).  This  direction  may  be  determined 
from  the  + and  — signs  on  the  connecting  leads.  Change 
the  connections  so  that  the  current  is  reversed  in  one  of  the 
coils  and  proceed  as  before  to  note  the  relative  directions 
of  current  and  lines  of  force  indicating  these  on  the  data 
sheet  by  means  of  small  arrows. 

Replace  the  coils  by  the  solenoid  and  repeat  the  perfor- 
mance, recording  the  directions  of  the  lines  of  force  about 
the  horizontal  cross  section  represented  on  the  data  sheet. 
Compare  this  field  with  that  of  the  bar  magnet, 
v.  Question. — 1. — What  are  some  of  the  properties  of 

lines  of  force,  illustrated  in  these  blue  prints? 

^ 2. — Draw  the  magnetic  field  of  force  about  a bar  magnet 
and  a compass  needle,  placed  as  shown  in  Fig.  42,  and  ex- 


162 


Manual  of  Experiments 


plain  the  effect  produced  upon  the  needle,  by  means  of  the 
properties  of  lines  of  force. 

3. — Draw  the  field  of  force  in  a horizontal  plane  through 
the  middle  of  a vertical  coil  of  wire  carrying  a current  of 
electricity  and  having  its  plane  in  the  magnetic  meridian 
with  a compass  needle  suspended  at  its  center,  (tangent 
galvanometer,  see  Fig.  43)  and  explain  the  effect  produced 
upon  the  needle,  with  the  aid  of  the  properties  of  lines  of 
force. 

32.  ACTION  BETWEEN  MAGNETIC  POLES 
Superposition  of  Magnetic  Fields 

To  study  the  law  of  action  between  magnetic  poles  and 
to  show  the  effect  of  superimposing  one  magnetic  field  upon 
another.  (D.  372,  380;  G.  716-719,  736;  K.  479,  485,  489- 
491;  W.  416,  421,427.) 

Apparatus. — Two  bar  magnets,  small  compass,  stand 
with  stirrup  for  suspending  magnets,  sheet  of  paper. 

Theory  and  Method. — The  law  of  action  between 
magnetic  poles  is  that  “'like  poles  repel  each  other;  unlike  poles 
attract  each  other. J ? This  law  is  readily  verified  by  suspend- 
ing one  of  the  bar  magnets  so  that  it  is  free  to  rotate  in  a 
horizontal  plane  about  an  axis  perpendicular  to  its  own,  and 
approaching  first  one  and  then  the  other  pole  of  a second 
magnet  to  the  same  pole  of  the  suspended  magnet.  A mag- 
net suspended  as  described  forms  a torsion  pendulum  and  it 
performs  harmonic  vibrations.  The  characteristic  of  such 
motions  is  that  the  acceleration  at  any  instant  is  proportional 
to  the  displacement  from  the  position  of  rest,  and  the  period 
of  the  vibrations  is 


T = 2 t y/  — 4>  / a 


In  General  Physics 


163 


where  “0 M is  the  angular  displacement  and  11  a11  the 
angular  acceleration. 

When  a magnet  of  length  11 V'  and  pole  strenght  “m” 
is  suspended  in  the  magnetic  field  of  strength  77,  making 
an  angle  0 with  the  direction  of  the  field,  the  force  acting 
on  each  pole  is  77m,  and  the  moment  arm  is  (I/2)  sin  0 so 
that  taking  both  poles  into  consideration,  the  moment  of 
the  couple  tending  to  turn  the  magnet  is 

2m77  {1/2)  sin<fr  = MH  sin  0 - 

✓ 

for  the  magnetic  moment  of  the  magnet,  717,  is  equal  to  ml. 
But  if  7 be  the  moment  of  inertia  of  the  magnet,  the  turning 
moment  acting  on  it  will  be  la , and  since  the  displacement 
and  acceleration  are  always  in  opposite  directions  0 and  a 
will  always  have  opposite  signs.  Therefore 

M 77  sin  0 = — 7 a - 

and 

- sin* /a  = 7/  (717  77) 


For  small  angles  of  vibration  sin<t>  may  be  considered  equal 
to  0 itself,  so  that 


- 0 / « = 7/  (717  77). 


Substituting  this  value  in  the  expression  for  the  period, 
we  have 


2 7 r 


/_/_ 

V 77717 


From  this  we  see  that  the  period  of  a vibrating  magnet 
is  inversely  proportional  to  the  square  root  of  77,  the 
strength  of  the  field  in  which  it  vibrates,  and  of  717,  its 
magnetic  moment. 


164 


Manual  of  Expekiments 


When  a second  magnet  is  brought  into  the  neighbor- 
hood of  the  suspended  one,  its  field  is  superimposed  upon 
that  of  the  earth,  and  the  strength  of  field  acting  on  the  latter 
magnet  is  decreased  or  increased  according  as  like  poles  or 
unlike  poles  are  placed  adjacent  to  each  other.  In  other 
words,  since  the  predominating  action  is  that  between  ad- 
jacent poles,  the  action  of  the  second  magnet  upon  the  first 
is,  in  the  second  case  the  same  as  that  of  the  earth,  in  the 
first  case  opposed  to  that  of  the  earth. 

If  both  magnets  be  suspended  together,  they  may  be 
considered  as  forming  a single  magnet,  its  magnetic  moment 
being  greater  or  less  than  that  of  either  alone. 

If  the  two  magnets  have  their  unlike  poles  in  juxtaposi- 
tion, and  have  nearly  the  same  pole  strength,  the  directive 
force  tending  to  keep  them  in  the  magnetic  meridian  is  very 
much  reduced.  Such  a pair  of  magnets  of  very  nearly  the 
same  strength,  suspended  with  opposite  poles  adjacent  is 
called  an  astatic  system  of  magnets,  and  is  very  sensitive  to 
influences  tending  to  turn  it  from  the  magnetic  meridian. 

Since  the  earth  is  a large  magnet  having  a magnetic 
north  pole  in  the  neighborhood  of  its  geographic  south 
pole  and  vice  versa,  any  magnet  placed  in  the  earth ’s  field 
sends  lines  of  force  to  the  earth ’s  south  magnetic  pole  and 
receives  lines  of  force  from  the  earth ’s  north  magnetic  pole. 

Unless  a magnet  be  perfectly  symmetrical  about  its 
axis  and  be  in  the  magnetic  meridian  with  its  north  seeking 
pole  towards  the  north,  the  field  of  force  surrounding  it 
will  be  distorted.  Any  mass  of  iron  in  the  earth ’s  field  and 
especially  vertical  iron  pipes  and  columns,  being  magnetic 
by  induction,  distort  the  earth ’s  field  in  their  neighborhood. 
The  distorted  fields  due  to  magnets  may  be  studied  by  plac- 
ing a small  compass  in  the  field  and  moving  it  continuously 
in  the  direction  in  which  it  points,  marking  the  path  followed 


i 


In  General  Physics 


165 


with  a pencil.  The  magnetic  intensity  at  any  point  of  this 
field  is  the  resultant  of  the  forces  exerted  by  the  poles  of 
the  magnet  and  those  of  the  earth  upon  a free  north  pole 
at  that  point.  By  Coulomb’s  law  these  forces  are  inversely 
proportional  to  the  square  of  the  distance  from  the  point 
to  each  pole.  Since  the  compass  needle  indicates  the  direc- 
tion of  the  resultant  force  and  is,  at  every  point  of  the  paths 
described,  tangent  to  the  path,  these  paths  represent  lines 
of  force. 

Directions. — Suspend  each  of  the  bar  magnets  with  its 
axis  in  a horizontal  plane  by  means  of  the  stirrup  and  fine 
wire,  (See  Fig.  39),  and  allow  it  to  vibrate  until  it  indicates 
clearly  which  is  the  north  seeking  pole.  The  north  seeking 
pole  should  be  marked  with  a file  or  pencil  mark. 


Action  Between  Poles. — Suspend  one  of  the  magnets  and 
bring  it  to  rest.  Then  approach  the  north  pole  of  the  other 
to  that  of  the  suspended  magnet  and  record  the  resulting 
action  between  them.  Bring  up  an  unlike  pole  and  record 
the  resulting  action. 


Effect  of  Strengthening  or  Weakening  the  Earth’s  Field. — 
Turn  the  magnet  from  its  position  of  rest  through  a small 


166 


Manual  of  Experiments 


angle  (about  30°)  and  allow  it  to  vibrate  about  a vertical 
axis  through  its  center  of  suspension  in  a horizontal  plane. 
Note  the  time  for  five  complete  vibrations  and  calculate 
the  mean  period.  Do  the  same  for  the  second  magnet. 
Next  place  the  first  magnet  on  the  table  near  the  second 
(suspended)  magnet,  like  poles  being  adjacent.  (See  Fig. 
40).  Let  the  suspended  magnet  vibrate  through  a small 
arc,  note  the  time  of  five  complete  vibrations  and  calculate 
the  new  period.  (A  second  trial  with  the  distance  between 
magnets  altered  may  be  necessary  to  show  the  change  in  the 
period.  Do  not  have  the  magnets  so  close  together  that 


< \ 
\ 

\ 

\ 


v 


Fig.  40 


the  suspended  magnet  reverses  its  direction.)  Repeat  with 
unlike  poles  adjacent. 

Astatic  System  of  .Magnets . — Suspend  both  magnets  to- 
gether, having  like  poles  contiguous  and  find  the  mean 
period.  Reverse  one  of  the  magnets  and  repeat.  (If  the 
reversed  magnet  has  the  stronger  pole  the  system  as  a whole 
will  reverse  itself.) 

Distortion  of  Earth’s  Field . — Place  a sheet  of  paper  with 
one  edge  parallel  to  that  of  the  table  and  lay  a bar  magnet 
at  the  middle  of  the  west  edge,  with  its  north  pole  pointing 


In  General  Physics 


167 


north.  Draw  an  outline  of  the  magnet  and  label  its  ends 
properly  + and  — . Place  a small  compass  at  the  middle 
of  the  + pole  and  move  it  about  a quarter  of  an  inch  in  the 
direction  in  which  the  needle  points  when  it  has  come  to 
rest,  marking  the  direction  of  the  needle  by  a short  line, 
1/8  inch  long,  behind  the  compass.  ' Continue  this  operation 
until  the  edge  of  the  paper  or  the  other  pole  has  been  reached. 
Starting  from  six  equidistant  points  about  one  side  of  each 
pole,  proceed  as  before.  (A  semi-circle  may  be  drawn  about 
each  pole  with  the  pole  as  center,  and  six  equidistant  points 
laid  off  on  each.  ) 

From  some  point  c on  one  of  the  lines  of  force  which 
extends  from  one  pole  to  the  other  and  near  both  poles , draw 
lines  in  the  direction  of  the  force  exerted  by  each  of  the  two 
poles  on  a free  north  pole  at  that  point,  and  on  these  lines 
lay  off  lengths  which  are  inversely  proportional  to  the  squares 


of  the  respective  distances  from  the  point  to  the  pole  exerting 
the  force.  With  these  lines  as  component  forces  draw  the 
resultant  force  and  note  its  position  with  respect  to  the  line 
of  force.  (See  Fig.  41.) 

On  the  reverse  side  of  the  paper  proceed  to  trace  the  lines 
of  force  as  before  but  with  the  magnet  in  the  center  of  the 
paper,  its  + end  west  and  its  — end  east  . 


168 


Manual  of  Experiments 


Make  a reduced  copy  of  these  fields  of  force  on  the  data 
sheet. 

Questions. — 1.  In  the  last  part  of  the  experiment  why 
would  it  not  be  satisfactory  to  find  the  resultant  force  at  a 
point  distant  from  the  magnet  by  this  method? 

'x  2. — Two  bar  magnets  when  suspended  as  an  astatic 
system  have  a period  of  a ( = 5.)  seconds,  but  when  the  weaker 
one  is  reversed  in  the  system  the  period  is  b ( = 3.)  seconds. 
If  both  have  the  same  dimensions  find  the  ratio  of  their 
magnetic  moments. 

^ 3. — A certain  magnetic  needle  has  a natural  period  of 
a ( = 4.)  seconds.  When  suspended  above  and  parallel  to 
a bar  magnet  lying  in  the  magnetic  meridian,  its  period  is 
b ( = 20.)  seconds.  How  many  vibrations  per  second  will 
it  make  if  the  bar  magnet  is  reversed? 

4. — At  a place  where  the  earth’s  magnetic  field  has  a 
horizontal  intensity  a ( = 0.2.)  units,  two  parallel  bar  magnets 
each  b ( = 12.)  cm.  long,  are  placed  with  their  axes  horizontal 
and  in  the  magnetic  meridian,  with  their  unlike  poles  oppo- 
site each  other  and  c ( = 16.)cm.  apart.  The  poles  pointing 
toward  the  south  are  d ( = + 50.)  and  e ( = — 25)  units  in 
strength  respectively.  Find  the  total  force  acting  on  a pole 
of  strength  / ( = +5)  units  at  a point  midway  between  the 
four  poles  of  the  magnets.  State  the  direction  of  the  force 
as  well  as  its  magnitude.  , 

33.  DETERMINATION  OF  H AND  M 

To  determine  the  horizontal  intensity  of  the  earth’s 
field  and  the  magnetic  moment  of  a magnet.  (D.  375-384; 
G.  723,  724,  736;  K.  491,  492,  501;  W.  423-428.) 


Apparatus. — Large  compass  box  with  attached  meter 


In  General  Physics 


169 


rod,  small  compass,  magnetometer  box  with  suspension, 
magnet  and  holder. 

Theory  and  Method. — If  a magnetic  needle  is  free  to 
rotate  about  a point  in  the  earth ’s  field  it  will  place  itself 
in  the  magnetic  meridian,  making  an  angle  with  the  hori- 
zontal, known  as  the  angle  of  inclination  or  dip.  If  the 
needle  is  free  to  rotate  in  the  horizontal  plane  only,  it  will 
be  acted  upon  by  the  horizontal  component  i7,  of  the  earth’s 
field  and  will  set  itself  in  the  magnetic  meridian.  By  the 
horizontal  intensity  of  the  earth’s  field  at  any  point  we  mean 
the  number  of  dynes  of  force  acting  in  the  horizontal  plane 
upon  a unit  north  magnetic  pole  at  that  point. 

The  period  of  vibration  of  a horizontally  suspended  mag- 
net about  its  center  is 


' T = 2ttV  I /MH 

(See  theory  of  experiment  32).  From  this  equation  we  get 


MH  = 


7T 


2 / 


y2 


We  have  also  a method  of  determining  M/H  and  are 
therefore  able  to  calculate  both  M and  H.  Let  a magnet 
whose  period  has  been  determined  be  placed  with  its  axis 
perpendicular  to  the  magnetic  meridian  and  in  a line  with 
the  center  of  a suspended  magnetic  needle,  the  distance 
between  their  centers  being  “a’  ^(See  Fig.  42).  The  earth ’s 
field,  H , and  that  of  the  magnet,  H'  will  be  approximately 
uniform  and  perpendicular  to  each  other  at  the  magnetic 
needle,  and  will  cause  it  to  be  deflected  through  an  angle 
( / >,  its  axis  lying  in  the  direction  of  their  resultant,  so  that 


tan<p  = H' / H 

Let  m be  the  pole  strength  and  l the  length  of  the  magnet. 


170 


Manual  of  Experiments 


By  Coulomb’s  law,  the  force  between  two  poles  of  strength 
m and  m'  at  a distance  r apart,  in  air,  is 


F = 


mm 


Then,  by  definition,  the  intensity  of  a field  due  to  a pole  of 
strength  m,  at  a distance  r from  the  pole  is  m/r2.  If  the 
suspended  magnet  is  quite  small  compared  with  l and  a , the 


-m 


\m 


Fig.  42 


approximate  distance  of  the  poles  +m  and  — m from  the 
poles  of  the  suspended  magnet  are  (a  — 1/2)  and  (a  + 1/2) 
respectively,  and  as  H'  is  the  field  due  to  both  +ra  and  —m, 


H'  = 


m 


m 


(a  - 1/ 2) 2 
Combining  these  terms  we  have 

2 mla 


(a  + 1/2)- 


H'  = 


2 Ma 


(a2  — /2/4)2  {a2  — /2/4 ): 


where  M is  the  magnetic  moment  of  the  magnet.  As  l is 
usually  small  compared  with  a,  the  terms  2a2l2/\  and  /4/16 
in  the  expansion  of  the  denominator  may  be  neglected  and 
we  have 


H'  = 


2 Ma  2 M 


a? 


In  General  Physics 


171 


Substituting  this  value  in  the  expression  for  tan<t> 

tan4>=  (2M)/(azH)  or 

M i 3 , 

~^zr  — — a6  tan  </> 

By  combining  this  ratio  with  the  expression  for  the  product 
of  H and  M so  that  first  one,  then  the  other  of  these  terms 
is  eliminated,  we  get  expressions  for  both  H and  M in  terms- 
of  measurable  quantities. 

Directions. — To  determine  MH. 

Place  the  magnetometer  box  with  the  glass  sides  facing 
north  and  south  respectively.  Suspend  the  magnet  in  the 
stirrup  so  that  it  lies  in  the  horizontal  plane  when  it  comes 
to  rest  in  the  magnetic  meridian.  Then  start  it  to  oscil- 
lating through  an  angle  of  about  10°,  preventing  any  motion 
of  the  center  of  the  magnet  by  means  of  the  wire  stop  at  the 
side  of  the  box.  Determine  the  period  of  the  motion  from 
the  time  required  for  ten  complete  oscillations.  Repeat  this 
determination  as  a check.  The  magnets  are  marked  with 
Roman  numerals.  Note  from  the  bulletin  board  the  mass, 
length  and  radius  of  the  magnet  you  used,  and  calculate  its 
moment  of  inertia  and  the  value  of  MH.  (See  Appendix 
for  formulae  of  moments  of  inertia). 

To  determine  M/H. 

The  large  compass  box  has  a small  magnetic  needle  with 
a long  aluminum  pointer  attached  perpendicularly  to  it, 
moving  over  a circular  scale  on  a glass  mirror.  A meter 
stick  is  fastened  symmetrically  under  the  center  of  the  box. 
The  compass  is  turned  until  the  needle  hangs  at  rest,  with 
the  pointer  in  line  with  the  meter  stick,  and  at  the  same  time 
the  leveling  screws  are  adjusted  so  that  the  pointer  may 
swing  freely  about  the  center  of  the  scale.  Be  sure  the  mag- 


172 


Manual  of  Expeeiments 


net  is  not  close  enough  to  affect  the  needle.  Note  on  the 
scale  the  position  of  the  ends  of  the  pointer.  To  avoid  errors 
due  to  parallax  in  taking  this  reading,  place  the  eye  vertically 
above  each  end,  so  that  the  pointer  is  seen  covering  its 
image  in  the  mirror.  Place  the  magnet  symmetrically  in 
the  holder  and  put  them  on  the  meter  stick  with  centers 
25  cm.  east  of  the  needle.  Note  the  direction  of  both  ends 
of  the  pointer.  Then  reverse  the  magnet  and  when  the 
needle  has  come  to  rest  note  the  deflection  of  both  ends  from 
their  original  position.  Repeat  for  25  cm.  west  of  needle. 
Do  the  same  for  26  and  27  cm.  Calculate  the  mean  de- 
flection in  each  case  and  the  values  of  the  tangent  of  this 
angle  times  the  distance  cubed.  From  the  average  of  these 
products  calculate  the  value  of  M/H , and  combining  with 
the  value  found  for  M77,  determine  both  M and  H. 

Questions. — 1.  Explain  how  the  following  arrange- 
ments of  the  compass  box  would  introduce  errors  in  the 
measurements  and  how  the  errors  were  eliminated  in  each 
case: 

a)  The  center  of  the  scale  did  not  coincide  with  the 
center  of  suspension? 

b)  The  center  of  the  meter  rod  was  not  under  the 
center  of  suspension? 

c)  The  poles  of  the  bar  magnet  were  not  equidistant 
from  the  center  of  the  magnet? 

^ 2.  Find  the  least  distance  between  magnet  and  needle 
in  the  above  experiment  for  which  the  approximate  formula 
H'  = 2M/as  can  be  used  with  a magnet  ^(  = 10)cm  long 
without  making  an  error  greater  than  b{  — 1.)%  in  the  value 
of  H'. 

^3.  A rectangular  bar  magnet  having  a length  a{  = 8.)cm., 
width  b{  — 2.)  cm.,  thickness  c(  = 1.)  cm,  and  mass  ^(  = 120). 


In  General  Physics 


173 


grams  makes  <?(  = 10)  vibrations  per  minute  in  a magnetic 
field  whose  strength  is  /(  = 0.2)  units.  What  is  its  pole 
strength?  (See  Appendix  IV  for  moments  of  inertia.) 

^4.  A bar  magnet  with  length  a(  = 10)  cm.  and  pole 
strength  £(  = 20)  units  lies  in  the  A position  of  Gauss  with 
respect  to  a small  compass  needle,  its  center  being  r(  = 100) 
cm.  east  of  the  needle,  its  N.  pole  toward  the  east.  A 
similar  magnet  with  length  d{  = 20)  cm.  and  pole  strength 
/(  = 5.)  units  lies  in  the  B position  of  Gauss,  north  of  the 
needle,  its  axis  east  and  west.  If  the  needle  is  not  deflected 
from  the  magnetic  meridian,  how  far  is  it  from  the  center 
of  the  second  magnet? 


34.  GALVANOMETERS 

To  compare  different  types  of  galvanometers  and  to 
study  the  law  governing  the  deflections  of  a tangent  gal- 
vanometer. (D.  430-439,  442,  443,  449;  G.  835-845,  850, 
856;  K.  600-603;  W.  476-480.) 

Apparatus. — Tangent  galvanometer,  astatic  galvano- 
meter, D4Arsonval  galvanometer,  resistance  box,  gravity 
cell,  reversing  switch,  five  connecting  wires. 

os** 

Theory  and  Description. — A galvometer  is  an  instru- 
ment for  measuring  the  strength  of  an  electric  current. 

By  strength  or  intensity  of  current  we  mean  the  rate  of 
flow  of  quantity  of  electricity.  The  quantity  of  electricity 
which  passes  through  any  cross  section  of  a conductor  in 
a given  time  is  therefore  equal  to  the  current  strength 
multiplied  by  the  time. 

The  absolute  unit  of  current  strength  in  the  C.G.  S.  system 
is  defined  as  that  current  which , when  flowing  through  a con - 


174  Manual  of  Experiments 

ductor  one  centimeter  long  and  bent  in  a circular  arc  of  one 
centimeter  radius , acts  with  a force  of  one  dyne  on  a unit 
magnetic  pole  at  the  center  of  the  arc.  The  practical  unit  of 
current  strength  is  called  the  ampere , and  is  one - tenth  as 
great  as  the  absolute  unit  of  current  strength. 

The  absolute  unit  of  quantity  of  electricity  is  defined  as 
the  quantity  of  electricity  which  passes  through  any  cross 
section  of  a conductor  when  one  absolute  unit  of  current  strength 
flows  for  one  second.  The  practical  unit  of  quantity  is  called 
the  conlomb  and  is  one  tenth  as  great  as  the  absolute  unit  of 
quantity.  The  conlomb  may  also  be  defined  a$  the  quantity 
of  electricity  which  flows  through  any  cross  section  of  a con- 
ductor when  a current  strength  of  one  ampere  flows  for  one 
second. 

The  essential  parts  of  an  ordinary  galvanometer  are  a 
coil  of  wire  through  which  the  current  flows,  and  a per- 
manent magnet,  either  one  of  these  being  fixed,  the  other 
being  suspended  so  that  it  is  free  to  rotate  in  the  magnetic 
field  of  force  of  the  first. 

The  tangent  galvanometer  has  a flat  circular  coil  of  wire, 
C,  fixed  in  position,  with  its  plane  in  the  magnetic  meridian 
so  that  when  a current  flows  through  it  the  magnetic  field 
of  force  is  at  right  angles  to  that  of  the  earth.  A small 
magnetic  needle,  A,  is  suspended  by  means  of  a silk  fibre 
•at  the  center  of  the  coil.  (See  Fig.  43).  When  a current  of 
electricity  passes  through  the  coil,  the  needle  will  be  de- 
flected from  its  north  and  south  direction  u;ntil  its  axis  lies 
in  the  direction  of  the  resultant  of  the  two  fields  of  force. 
If  one  absolute  unit  of  current  flows  through  our  galvano- 
meter coil  having  n turns  of  radius  r,  since  the  force  varies 
as  the  length  of  wire  and  inversely  as  the  square  of  the 
distance  r,  it  will  act  on  a unit  magnetic  pole  at  the  cen- 
ter with  a force  (2irrn)/(r2)  = (2Trn)/r  dynes.  This  force 


In  General  Physics 


175 


Fig.  43 


(2irn)/r  dynes  is  called  the  galvanometer  constant,  G.  The 
strength  of  field,  F,  at  the  center  due  to  a current  of  / abso- 
lute units  in  the  coil  will  be  IG  dynes,  and  will  be  practically 
uniform  throughout  the  space  in  which  the  small  needle 
moves.  If  the  needle  has  a magnetic  moment  M and  is 
deflected  through  an  angle  </>  the  moment  of  the  deflecting 
couple  will  be  IGMcosfy  and  the  moment  of  the  balancing- 
couple  due  to  the  earth’s  field  of  strength  H will  be  HM 
sincf).  These  moments  must  be  equal  when  the  needle  is 
at  rest  or, 

IGMcos.4>  = HMsin.cf) 


and 


H Hr 

— tan.  6 = 

G 2irn 


tan.4> 


176 


Manual  of  Experiments 


H and  G are  constants  for  a given  galvanometer  in  a given 
position  and  their  ratio  K is  called  the  reduction  factor  of 
the  galvanometer,  the  current  strength  in  absolute  units 
being  K times  the  tangent  of  the  angle  of  deflection.  Since 
the  practical  unit  of  current  strength  is  one  tenth  as  great 
as  the  absolute  unit,  the  number  of  practical  units  C will  be 
ten  times  as  great,  or  the  current  strength  in  amperes  is 

10#  10  Hr 

C = tan  d>—  — ■ — tan  <b 

G 2irn 

If  the  plane  of  the  coil  C is  not  in  the  magnetic  meridian 
but  makes  an  angle  6 with  it,  the  field  of  the  coil,  F,  and  the 
earth’s  field,  #,  will  form  an  angle  of  90°  =*=  6 instead  of 
being  at  right  angles.  (See  Fig.  44.)  F will  have  a com- 
ponent F sin  6 in  the  same  direction  as  H or  opposite  to  it, 
and  a component  F cos  6 at  right  angles  to  H.  The  rec- 
tangular components  of  the  resultant  force,  R,  which  pro- 
duces the  deflection  of  the  suspended  magnetic  needle,  are 
then  H =*=  F sin  0 and  F cos  6 , the  + or  — sign  depending 
upon  the  direction  of  F relative  to  that  of  H,  that  is  upon 
the  direction  of  the  current.  In  other  words,  if  the  current 
flows  in  one  direction  the  field  Ri  will  be  the  resultant  of 
F cos  6 srndH  + F sin  0,  where  as  if  the  current  is  reversed, 
the  field  R will  be  the  resultant  of  F cos  6 and  Ft  — F sin  6. 

'The  resulting  deflections  will  be  unequal.  (Both  cases  are 
shown  in  figure  44,  the  respective  deflections  being  marked 
</>'  and  0.)  When  the  needle  is  deflected  through  an  angle 
</>  and  comes  to  rest,  the  moments  of  the  deflecting  and 
the  restoring  couple  will  be  equal.  Hence 

(Ft  =*=  FsinO)  Msin4>  = F cos  6 Mcos<j>} 


and 


In  General  Physics 


177 


tan  0 = 


F cos  6 
H =±=  F sm  0 


In  order  that  the  deflections  may  be  equal  for  direct  and 
reversed  current,  however,  the  plane  of  the  coil  must  be  in 
the  magnetic  meridian,  or  6 = o.  Under  this  condition 
sin  0 — o and  cos  0=1,  hence  the  previous  equation  becomes 

F IG  2t nl 

tan  0=  — = — — = — - — — 

- H H Hr 


and 


178 


Manual  of  Experiments 


H Hr 

I = — tan  6= tan  d> 

G 2tt  n 

as  before. 

In  order  to  read  the  deflections  of  the  needle  (of  the  tan- 
gent galvanometer  to  be  used)  a long  aluminum  pointer,  P, 
fastened  at  right  angles  to  it,  moves  over  a circular  scale 
etched  on  glass.  The  lower  side  of  the  glass  is  silvered, 
forming  a mirror  so  that  errors  due  to  parallax  may  be 
avoided  by  reading  the  position  of  the  pointer  when  it  and 
its  image  are  in  the  same  lind  of  vision.  It  can  be  shown 
that  the  results  are  least  affected  by  errors  in  these  readings 
when  the  angle  of  deflection  is  about  45°.  The  galvano-  - 
meters  to  be  used  have  three  coils  the  number  of  turns  in 
each  being  different.  That  particular  coil  should  be  used 
in  general,  which  with  the  available  current  strength  gives 
a deflection  of  from  30°  to  60°. 

From  the  above  equation  for  the  current  strength  it  is 
seen  that  the  sensitiveness  of  such  a galvanometer  to  weak 
currents  is  increased  by  increasing  the  number  of  turns  of 
wire  in  the  coil  or  by  decreasing  the  radius  of  the  coil  or  the 
directive  force  of  the  earth  upon  the  suspended  magnet. 
In  the  astatic  galvanometer  all  three  of  these  methods  for 
increasing  the  sensibility  are  used,  The  needle  system  is 
astatic  (see  exp.  32),  the  number  of  turns  of  wire  is  large, 
and  the  coil  is  much  smaller  and  has  the  shape  of  a flat  oval, 
the  conductors  lying  closer  to  the  needles,  so  that  they  are 
in  a stronger  deflecting  field.  In  some  galvanometers  each 
of  the  two  needles  is  at  the  center  of  a coil,  the  two  coils 
being  wound  in  opposite  directions,  so  that  the  deflecting 
force  acting  on  the  needles  is  in  the  same  direction  for  both. 
In  others  the  directive  force  of  the  earth  is  weakened  by 
adjusting  a permanent  magnet  above  the  galvanometer  in 
such  a position  that  it  opposes  the  earth’s  field. 


In  General  Physics 


179 


Both  the  tangent  and  the  astatic  galvanometers  are  pro- 
tected from  air  currents  by  glass  covers,  but  both  are  subject 
to  the  disturbing  influences  of  surrounding  magnetic  fields. 

In  the  D’ Arson  val  galvanometer,  the  small  rectangular 
coil  which  carries  the  current  is  suspended  by  means  of  a 
phosphor-bronze  or  steel  strip  between  the  poles  of  a fixed 
permanent  U shaped  magnet.  A fixed  cylinder  of  soft  iron 
projecting  within  the  coil,  serves  to  make  the  magnetic  field 
in  which  the  coil  moves,  stronger  and  more  uniform  so  that 
within  certain  limits  the  deflections  are  directly  proportional 
to  the  current  flowing  in  the  coil.  The  magnet  is  sufficiently 
strong  to  make  the  influence  of  the  earth  and  of  neighboring 
magnetic  fields  negligible,  and  the  galvanometer  does  not 
have  to  be  fixed  in  any  definite  position.  In  order  to  observe 
the  small  deflections  of  the  coil  accurately,  a small  plane 
mirror  is  fastened  to  it.  A millimeter  scale  is  fixed  at  about 
half  a meter  in  front  of  the  mirror  and  its  image  in  the 
mirror  is  observed  by  means  of  a telescope  fastened  to  the 
scale.  The  apparent  deflection  of  the  image  of  the  scale 
as  observed  through  the  telescope  is  twice  as  great  as  the 
actual  deflection  of  the  coil  and  mirror.  (See  Fig.  22). 
The  telescope  may  be  dispensed  with,  if  the  plane  mirror 
is  replaced  by  a concave  one.  In  this  case  the  scale  is 
fastened  to  a plate  of  ground  glass  or  other  semi-transparent 
substance,  and  an  illuminated  slit  is  placed  at  its  center. 
The  distance  between  scale  and  mirror  is  adjusted  until 
the  image  of  the  slit  or  of  the  lamp  filament  is  sharply 
focussed  on  the  ground  glass,  so  that  the  deflections  of 
the  image  may  be  observed. 

Either  of  the  above  two  types  of  galvanometers  may  be 
designed  for  use  in  three  different  kinds  of  measurements. 

a)  We  may  wish  to  measure  a current  of  definite 
strength  which  produces  a steady  deflection  of  the  suspended 


180 


Manual  of  Experiments 


coil  or  needle.  The  tangent  and  the  D’Arsonval  galvano- 
meter are  frequently  used  for  this  purpose.  The  galvano- 
meter should  be  aperiodic  or  dead-beat;  that  is,  its  suspended 
needle  or  magnet  should  come  to  rest  quickly,  without 
vibrating  back  and  forth.  In  the  first  type  of  galvanometer 
described,  this  damping  is  produced  by  surrounding  the 
needle  (usually  of  an  inverted  U shape)  with  a mass  of 
copper.  In  the  D’Arsonval  type  the  damping  is  produced 
by  winding  the  coil  on  a metal  frame  or  placing  it  in  a metal 
tube.  In  either  type  eddy  currents  are  induced  in  the  solid 
metal  and  prevent  vibrations.  (See  Exp.  43). 

b)  We  may  wish  to  detect  the  presence  of  very  small 
currents,  in  which  case  a very  sensitive  galvanometer  such 
as  an  astatic  or  a D’Arsonval  is  needed.  This  is  the  case  in 
all  measurements  where  a null  method  is  used;  that  is, 
where  the  apparatus  is  to  be  adjusted  until  no  current 
flows.  (See  Exp.  36,  37,  38,  etc.) 

c)  We  may  wish  to  measure,  not  the  strength  of  a 
current,  but  the  total  quantity  of  electricity  flowing  through 
the  galvanometer  during  a short  interval  of  time,  for  in- 
stance, the  discharge  of  a condenser,  or  an  induced  current. 
(See  Exp.  43.)  A galvanometer  adapted  to  such  measure- 
ments is  called  a ballistic  galvanometer.  It  should  have  as 
little  damping  as  possible,  and  a long  period  of  vibration 
so  that  the  needle  or  coil  does  not  move  appreciably  from  its 
position  of  rest,  before  all  of  the  electricity  has  passed 
through  the  coil.  Under  these  conditions  the  quantity  of 
electricity  is  proportional  to  the  sine  of  half  the  angle  of  the 
first  swing  (or  throw.) 

The  resistance  box  contains  several  coils  of  wire  of  dif- 
ferent lengths  and  cross  sections,  each  coil  having  a definite 
electrical  resistance  of  a whole  or  fractional  number  of  ohms. 
(See  Fig.  45.)  The  coils  are  wound  non -inductively  by  re- 


In  General  Physics 


181 


turning  the  wire  upon  its  own  path,  and  each  end  of  a coil 
is  connected  to  a separate  brass  block  on  the  upper  surface 
of  the  box.  Two  coils  are  connected  to  the  same  block  in 


such  a way  that  a current  of  electricity  may  be  passed  suc- 
cessively from  one  coil  into  the  other,  and  the  resistance  of 
the  circuit  may  be  varied  by  including  more  or  less  of  these 
coils.  But  when  a brass  plug  is  inserted  between  two  blocks, 
the  current  will  pass  partly  through  the  coil,  and  partly 
through  the  plug.  (See  Fig.  46.)  The  cross  section  of 
plugs  and  blocks  is  made  so  large  that  their  combined  re- 
sistance in  parallel  with  the  coil  is  negligible  (See  Exp.  36) 
and  for  all  practical  purposes  that  coil  is  cut  out  of  the  circuit. 
The  resistance  of  each  coil,  in  ohms,  is  marked  beside  the 
gap  between  the  blocks  to  which  the  ends  of  the  coil  are 
attached.  Unless  there  is  very  good  contact  between  the 
plugs  and  blocks,  the  resistance  to  the  flow  of  current 
through  these  surfaces  of  contact  will  be  appreciable  and 
may  cause  considerable  error  in  the  total  resistance. 


182 


Manual  of  Experiments 


It  is  advisable  to  use  a few  large  resistance  coils  rather 
than  a large  number  of  small  ones. 


Directions.— Adjust  the  level  of  the  tangent  galvano- 
meter so  that  the  pointer  swings  freely  over  the  whole  scale 
and  turn  the  galvanometer  until  the  plane  of  the  coil  is  in 
line  with  the  magnetic  needle  when  at  rest.  Remove  the 


W W A/j 


5lct  ion  on  RB 


Section  of  Rtsismuct  Box 

WIT  H 4 OHMS  IN  CIRCUIT 

fliro w*  Show  path  of  cvrrtnt 


Fig.  46 

short  circuiting  coil  from  the  gravity  cell  and  connect  the 
apparatus  as  shown  in  the  diagram,  Fig.  47,  G being  the 
galvanometer,  B the  galvanic  cell,  i?  the  resistance  box,  and 
S the  switch. 

Note  the  position  of  both  ends  of  the  pointer  when  the 
switch  is  open  in  which  case  the  resistance  of  the  circuit 
may  be  considered  infinitely  great  for  all  practical  purposes. 
Close  the  switch  and  when  the  needle  comes  to  rest  again 
note  the  position  of  the  pointer.  Then  reverse  the  current 
and  note  the  position  of  the  pointer.  If  the  mean  deflection 
is  not  the  same  in  both  cases  it  means  that  the  plane  of  the 
coil  is  not  in  the  magnetic  meridian  and  it  must  be  turned/ 
toward  the  magnet  (not  the  pointer)  in  the  position  of 
smaller  deflections,  until  the  deflections  become  equal  for 
reversed  currents.  The  box  resistance  during  this  trial  has 
been  zero,  and  when  the  deflections  are  equal  they  should 


In  General  Physics 


183 


\ 

be  recorded  for  both  ends  of  the  pointer  and  for  both  direct 
and  reversed  currents.  The  corresponding  deflections  for 
1,  2,  5,  and  10  ohms  of  box  resistance  in  the  circuit  should  be 
observed  and  the  mean  deflection  calculated.  This  box 
resistance  is  only  a part  of  the  total  resistance  of  the  circuit,- 
the  latter  including  the  resistance  of  cell,  galvanometer,  and 
connecting  wires.  To  determine  this  resistance  exclusive 
of  the  box  resistance,  plot  a curve  having  for  ordinates  the 
cotangents  of  the  angles  of  deflection  and  for  abscissae,  the 
corresponding  box  resistances.  The  intercept  of  this  curve 
on  the  axis  of  abscissae  between  the  origin  and  curve  gives 
the  required  resistance  to  the  same  scale  as  the  plotted  box 
resistance.  Calculate  the  total  resistance  in  each  case.  • 


According  to  Ohm’s  law,  the  current  strength  in  the  cir- 
cuit equals  the  electromotive  force  (e.  m.  /.)  in  the  circuit 
divided  by  its  total  resistance.  (See  Exp.  36.)  Note  the 
e.  m.  /.  of  the  cell,  and  calculate  the  current  strength  and 
the  reduction  factor  K of  the  galvanometer  in  each  case. 

Examine  one  of  the  D’Arsonval  galvanometers  on  the 
wall  and  include  a sketch  of  the  essential  parts  in  your 
written  report.  Do  the  same  for  the  astatic  galvanometer.  * 
Questions — 1. — a)  With  the  data  obtained  in  the 
above  experiment  plot  a curve  between  corresponding 
values  of  current  strength  (as  abscissae)  and  galvanometer 
deflections. 


184 


Manual  of  Experiments 


b)  On  the  same  sheet  of  coordinate  paper  plot  a second 
curve  between  corresponding  values  of  current  strength 
and  tangents  of  the  angles  of  deflection. 

2.  — Explain  why  the  intercept  of  the  above  curve  gives 
the  resistance  of  the  circuit  exclusive  of  box  resistance. 

3.  — a)  What  errors  due  to  the  lack  of  symmetry  in  the 
apparatus,  are  eliminated  by  the  method  of  taking  deflec- 
tions described  above. 

b)  Would  the  readings  be  affected  if  the  pointer  were 
bent,  or  if  it  were  not  fixed  at  right  angles  to  the  magnetic 
needle?  Give  reasons  for  your  answers. 

4.  — The  coil  of  a tangent  galvanometer  with  a steady 
I current  flowing  through  it  is  placed  at  right  angles  to  the 

magnetic  meridian.  The  needle  when  set  in  vibration  makes 
a(  = 5)  oscillations  per  second,  but  when  the  current  is 
reversed  it>makes  b(  = 3)  oscillations  per  second. 

a)  How  does  the  field  of  the  coil  at  its  center  compare 
with  that  of  the  earth? 

b ) What  will  be  the  deflection  of  the  needle  if  the  plane 
of  the  coil  is  placed  in  the  magnetic  meridian? 

35.  OHM’S  LAW  AND  POTENTIAL  DROP. 

To  test  Ohm’s  law  in  a simple  circuit,  and  to  show  that 
the  potential  drop  in  a conductor  is  proportional  to  the 
resistance  of  the  conductor.  (D.  pp.  395,  403,  411;  G.  pp. 
866-870,  953;  K.  pp.  415,  433,  434.) 

Apparatus.— D’Arsonval  galvanometer  with  shunt 
box,  resistance  box  (10,000  ohms),  high  resistance  wire 
with  sliding  contact,  gravity  cell,  reversing  switch,  and 
seven  connecting  wires. 

Theory  and  Description. — We  may  define  the  elec- 
trical potential  at  a point  as  the  work  done  in  bringing  unit 


In  General  Physics 


185 


positive  quantity  of  electricity  from  an  infinite  distance  up  to 
the  point  in  question.  It  is  shown  in  books  on  advanced 
physics  that  the  work  required  to  bring  unit  quantity  of 
electricity  from  an  infinite  distance  to  any  point  is  the  same 
whatever  path  may  be  followed.  In  bringing  this  quantity 
up  to  a second  point  we  may  therefore  follow  a path  which 
passes  through  the  first  point.  It  follows  that  the  difference 
of  potential  between  two  points  in  an  electric  circuit  is  the  work 
done  upon  unit  quantity  of  positive  electricity  in  bringing  it 
from  the  point  of  lower  potential  to  the  point  of  higher  po- 
tential; or  it  is  the  work  done  by  unit  quantity  of  positive 
electricity  as  it  flows  from  the  point  of  higher  potential  to  the 
point  of  lower  potential.  A “current”  shall  be  understood 
to  mean  a flow  of  positive  electricity.  (Unit  quantity  of  elec- 
tricity and  unit  current  strength  were  defined  in  experiment 
34.) 

Unless  there  is  a difference  of  potential  between  two 
points  of  a circuit  no  current  can  flow  between  them,  and 
positive  electricity  always  flows  from  a point  of  higher  to 
a point  of  lower  potential.  The  difference  of  potential 
between  two  points  is  therefore  often  spoken  of  as  the 
potential  drop  between  them. 

When  a current  of  electricity  flows  in  a simple  closed 
circuit  consisting  of  a source  of  electricity  (such  as  a gal- 
vanic cell  or  dynamo)  and  a conductor  which  connects 
the  terminals  of  the  source,  the  current  after  flowing  from 
one  terminal  through  the  external  conductor  must  flow 
back  to  the  first  terminal  through  the  conductors  within 
the  source,  (the  plates  and  liquid  of  the  cell,  or  the  wire 
coils  of  the  dynamo  armature),  and  there  must  be  a potential 
drop  within  the  source  as  well  as  through  the  external 
circuit.  The  electromotive  force  of  the  source  may  be 
defined  as  the  sum  of  the  external  and  internal  drops  of 


186 


Manual  of  Experiments 


potential  in  the  circuit;  or  in  general  the  electromotive  force 
in  a circuit  is  the  total  drop  of  potential  ( or  the  sum  of  all  the 
potential  differences ) met  with  in  passing  from  one  terminal 
of  the  source  of  electricity  through  the  complete  circuit  and 
hack  to  the  same  terminal. 

The  absolute  unit  of  potential  difference  or  electromotive 
force , in  the  C.  G.  S.  system , exists  between  two  points  in  a 
circuit  when  one  erg  of  work  must  be  done  in  order  to  move 
unit  quantity  of  positive  . electricity  from  the  point  of  lower 
to  the  point  of  higher  potential.  The  practical  unit  of  potential 
difference  or  electromotive  force  is  called  the  volt  and  is 

c 

108  times  as  large  as  the  absolute  unit.  The  volt  may  also 
be  defined  as  the  potential  difference  existing  between  two 
points  in  a circuit  when  one  joule  (107  ergs)  of  work  is  done 
by  one  conlomb  of  electricity  ( one  ampere  flowing  for  one 
second)  as  it  flows  from  the  point  of  higher  potential  to  the 
point  of  lower  potential. 

Ohm’s  law  states  that  when  a constant  electromotive  force 
is  impressed  upon  a closed  circuit , there  is  a constant  ratio 
between  the  electromotive  force  and  the  resulting  current  strength. 
This  constant  ratio  is  called  the  resistance  of  the  circuit. 
Ohm’s  law  is  more  commonly  stated  in  this  form:  The 

strength  (/)  of  the  electric  current  flowing  in  a closed 
circuit  equals  the  total  electromotive  force  (E)  in  the  circuit 
divided  by  its  total  resistance  (R),  or  in  symbols 


E 


The  total  electromotive  force  is  the  summation  of  the  e.  m. 
f.’s  of  all  sources  of  electricity  in  the  circuit,  each  with 
its  proper  sign.  (See  Exp.  38.)  The  total  resistance  of 
the  circuit  is  the  summation  of  the  resistances  of  all  the 
conductors  in  the  circuit,  the  method  of  summation  depend- 


In  General  Physics 


187 


mg  upon  the  manner  in  which  the  conductors  are  connected. 
(See  Exps.  36  and  37.) 

The  absolute  unit  of  resistance  in  the  C.  G.  S.  system  is 
the  resistance  of  a conductor  in  which  a current  strength  of 
one  absolute  unit  will  flow  when  the  absolute  unit  of  potential 
difference  exists  between  its  ends.  The  practical  unit  of 
resistance  is  called  the  ohm  and  is  109  times  as  large  as  the 
absolute  unit.  The  ohm  may  also  be  defined  as  the  re- 
sistance of  a conductor  in  which  a current  strength  of  one  ampere 
flows  when  one  volt  of  potential  difference  exists  between  its 
ends.  . . . 

Ohm’s  law  is  true  not  only  for  any  closed  circuit  as  a 
whole,  upon  which  a constant  e.  m.  f.  is  impressed,  but  it 
holds  for  any  part  of  the  circuit  as  well.  The  potential 
drop  between  any  two  points  of  the  circuit  must  equal 
the  resistance  between  these  two  points  multiplied  by 
the  strength  of  current  flowing  between  them,  and  the 
sum  of  all  the  potential  drops  met  with  in  passing  completely 
around  the  circuit,  must  equal  the  total  electromotive 
force  in  the  circuit. 

In  order  to  test  Ohm’s  law  we  may  connect  a source 
of  constant  e.  m.  f.  such  as  a gravity  or  storage  cell  in  a 
circuit  containing  a box  with  variable  resistance,  and  a 
galvanometer  arranged  to  measure  the  current  strength 
in  the  circuit.  (See  Exp.  39.)  If  the  box  resistance  is 
made  very  large  in  comparison  with  that  of  the  cell  and 
galvanometer  circuit,  the  latter  may  be  neglected.  On 
changing  the  box  resistance  the  current  will  change  in 
such  a way  that  the  product  of  current  strength  and  box 
resistance  will  remain  equal  to  the  electromotive  force. 
When  the  galvanometer  is  arranged  so  that  its  deflections 
are  directly  proportional  to  the  current  strength  producing 


188 


Manual  of  Experiments 


them,  the  product  of  box  resistance  and  corresponding 
deflections  will  be  constant  also. 

To  show  that  the  potential  drop  through  different  parts 
of  a circuit  is  directly  proportional  to  the  resistance  of 
these  parts  we  may  connect  a long  uniform  high  resistance 
wire  in  circuit  with  a source  of  constant  current  (gravity 
or  storage  cell),  and  arrange  a galvanometer  to  show  the 
potential  drops  through  different  lengths  of  the  wire.  (See 
Exp.  39.)  The  ratio  of  the  potential  drop  through  any 
section  of  the  wire  and  the  resistance  of  this  section  will 
be  constant.  Since  the  deflections  of  the  galvanometer 
are  directly  proportional  to  the  potential  difference  measured 
and  since  the  resistance  of  any  section  of  a wire  of  homo- 
geneous material  and  uniform  cross-section  is  directly 
proportional  to  the  length  of  this  section,  the  ratio  of  de- 
flections and  corresponding  lengths  will  be  constant  also. 

Directions. — a)  To  show  that  for  a constant  e.  m.  /, 
in  circuit  the  current  is  inversely  proportional  to  the  resistance . 
Connect  the  apparatus  as  shown  in  figure  47,  where  B is 
a gravity  (or  storage)  cell,  R a resistance  box  with  at  least 
10,000  ohms,  and  G a D’Arsonval  galvanometer  with  a 1 
ohm  shunt  (See  Exp.  36.),  connected  in  the  circuit  by  means 
of  a reversing  switch.  The  deflections  of  the  galvanometer, 
if  they  are  not  too  large,  will  be  proportional  to  the  current 
strength  in  the  circuit.  Remove  the  necessary  plugs  from 
the  resistance  box  to  insert  10,000  ohms  in  the  circuit. 
Close  the  switch  and  note  the  deflection  of  the  galvano- 
meter. Reverse  the  current  by  means  of  the  switch  and 
again  note  the  deflection.  Calculate  the  mean  of  the  two 
deflections  and  the  product  of  this  mean  and  the  box  re- 
sistance. Change  the  resistance  from  10,000  to  3,000  ohms 
in  steps  of  1,000  ohms,  and  note  the  deflections  for  direct 


In  General  Physics 


189 


and  reversed  current  in  each  case.  Calculate  the  products 
of  box  resistance  and  corresponding  mean  deflection  for 
each  case.  Compare  these  products  and  see  that  they 
are  practically  constant. 

b)  To  show  that  for  a constant  current  flowing  in  a circuit 
the  potential  difference  through  any  section  of  the  circuit  is 
directly  proportional  to  the  resistance  of  this  section.  Connect 
the  apparatus  as  shown  in  figure  56,  where  B is  a gravity 
(or  storage)  cell  connected  through  a reversing  switch  to 
the  ends  of  a high  resistance  wire  R.  This  wire  is  of  homo- 
geneous material  and  uniform  cross  section,  10  meters  long, 
and  is  stretched  back  and  forth  in  one  meter  lengths  on  a 
board  (See  Fig.  53.)  The  D’Arsonval  galvanometer  G 
with  a 10  ohm  shunt  S is  connected  through  a 10,000  ohm 
resistance  Rx  to  the  end  D of  the  long  wire  R.  The  other 
terminal  of  the  galvanometer  is  connected  to  a sliding 
contact  which  may  be  placed  at  any  point  on  the  long 
wire.  The  deflections  of  the  galvanometer  when  so  ar- 
ranged, are  proportional  to  the  potential  drop  through  the 
section  of  wire  across  which  it  is  connected.  Place  the 
sliding  contact  at  50  cm  from  the  end  D of  the  wire  and 
note  the  deflections  of  the  galvanometer  when  the  switch 
is  closed,  first  with  the  current  flowing  in  one  direction, 
then  with  the  current  reversed.  Calculate  the  mean 
deflection  and  the  ratio  of  this  mean  to  the  length  of  wire 
(50  cm.)  Do  the  same  with  the  sliding  contact  at  250, 
450,  650,  850  and  1000  cm  from  D respectively  and  cal- 
culate in  each  case  the  ratio  of  the  mean  deflection  to  the 
corresponding  length  of  wire.  Compare  these  ratios  and 
see  that  they  are  practically  constant. 

Questions,  etc. — 1. — a)  What  kind  of  curve  would 
you  obtain  if  you  plotted  total  resistance  (in  part  a)  of  the 


190 


Manual  of  Experiments 


above  experiment)  as  abscissae  and  corresponding  de- 
flections as  ordinates. 

b)  Plot  a curve  having  lengths  of  wire  (in  part  b above) 
as  abscissae  and  corresponding  deflections  as  ordinates. 

2.  — A storage  battery  with  an  internal  resistance  #(  = 0.5) 
ohms  sends  a current  £(  = 1.5)  amperes  thru  a resistance 
coil  of  c(  = 2.3)  ohms  in  series  with  an  incandescent  lamp 
whose  resistance  is  ^(  = 1.8)  ohms.  Find  the  potential 
drop  in  the  lamp  and  the  e.  m.  /.  of  the  battery. 

3.  — If  in  part  #)  of  the  above  experiment  the  products 
obtained  were  used  in  calculating  the  e.  m.  /.  of  the  cell, 
the  deflections  being  #(  = 1.)  mm.  per  £(  = 0.001)  ampere, 
the  resistance  of  the  cell  c(  = 10)  ohms  and  the  combined 
resistance  of  galvanometer  and  shunt  J(  = 0.98)  ohm, 
find  the  % error  in  the  calculated  e.  m.  f.  for  a box  resistance 
of  ^(  = 3000)  ohms. 

4.  — A battery  having  an  e.  m.  /.  #(  = 5.6)  volts  and 
internal  resistance  £(  = 2.)  ohms  sends  current  through  a 
wire  of  length  c(  = 20)  meters  and  resistance  ^(  = 38)  ohms. 
What  is  the  potential  drop  through  <?(  = 5)  meters  of  the 
wire? 

36.  THE  WHEATSTONE  SLIDE  WIRE  BRIDGE. 

To  determine  the  resistance  of  several  coils  of  wire, 
separately  and  in  combination,  by  means  of  the  Wheatstone 
bridge,  and  to  study  the  laws  for  determining  the  total 
resistance  of  various  combinations  of  separate  resistances. 
(D.  442,  443,  449-452,  456;  G.  853-856,  922,  923;  K.  642- 
645,  651,  652;  W.  480,  485-488.) 

Apparatus. — Wheatstone  bridge,  D’Arsonval  galvano- 
meter, galvanic  cell,  contact  key,  sliding  contact,  two  re- 
sistance boxes  with  known  and  unknown  resistance  coils 
respectively,  and  nine  connecting  wires. 


In  General  Physics 


191 


Description  and  Theory. — The  slide  wire  bridge  for 
comparing  resistance?  consists  of  a wire  w,  usually  one  meter 
long,  of  uniform  high  resistance,  stretched  on  a board  and 
soldered  at  the  ends  ( a and  b )to  two  heavy  copper  angle 
bars.  (See  Fig.  48).  A third  straight  bar  is  fastened  to  the 
board  so  that  there  is  a gap  between  it  and  each  angle  bar. 
A resistance  box  R is  connected  to  the  bars  at  one  of  these 
gaps,  the  unknown  resistance  X which  is  to  be  determined, 
is  connected  at  the  other  gap.  At  a point  midway  between 
the  two  a sensitive  galvanometer  G is  joined  to  the  straight 


bar  and  to  a sliding  contact  key  S which  may  be  moved 
along  the  wire  w.  A galvanic  cell  B is  connected  to  the  two 
angle  bars  at  c and  ^ with  a contact  key  K in  the  circuit. 
The  resistance  boxes  and  bars  together  with  the  wire  w form 
a divided  circuit  so  that  when  the  key  K is  closed  a current 
of  electricity  flowing  from  B to  c would  divide,  part  of  it 
flowing  through  w and  the  remainder  flowing  through  R 
and  X and  thence  from  e back  to  the  cell.  If  the  key  S be 
closed  there  will  in  general  be  a deflection  of  the  galvano- 
meter, but  on  moving  the  sliding  contact  along  the  wire  w 
a point  is  reached  where,  when  K and  S are  closed,  there  is 


192 


Manual  of  Experiments 


no  deflection  of  the  galvanometer,  which  indicates  that  there 
is  no  current  flowing  through  it. 

Ohm’s  law  holds  for  any  part  of  a circuit  as  well  as  for 
the  whole  circuit  (See  Exp.  35)  and  according  to  this  law 

Le  electromotive  force' 


the  potential  jfrop  (instead  oj 
through  any  part  of  the  circuit  equals  the  current  strength 
times  the  resistance  of  that  part  of  the  circuit.  If  there  is 
no  current  through  the  galvanometer  circuit,  there  can  be 
no  potential  drop  through  it,  or  in  other  words,  the  points 
d and  S are  at  the  same  potential.  Now  let  the  total  length 
of  wire  be  /,  let  the  length  aS  be  -x  and  its  resistance  be  P, 
then  the  length  of  bS  will  be  l — x and  its  resistance  say  Q . 
Further  let  the  current  strength  in  R be  I'  and  that  in  w 
be  I".  Since  no  current  flows  off  through  the  galvanometer 
circuit,  the  current  in  X will  be  the  same  as  that  in  R.  Then 
if  d and  S are  at  the  same  potential,  the  potential  drop  from 
c to  d must  equal  that  from  c to  S and  the  drop  from  d to  e 
must  equal  that  from  S to  e. 

That  is 

RI'-QI"  and  XI'  = PI" 


Dividing  the  second  equation  by  the  first  we  have 

XIf  PI"  P 

_ whence  X = — R. 

RI'  QI"  Q 


But  the  resistance  of  a conductor  of  uniform  cross  sec- 
tion and  homogeneous  material  is  directly  proportional  to 
its  length  (See  Exp.  37),  hence  the  ration  P/Q  equals  the 
ratio  x/(l  — x),  so  that 


v 

X = R. 

l—x 


The  unknown  resistance  may  therefore  be  calculated  from 
the  measurable  lengths  of  wire  and  the  known  resistance, 
neglecting  that  of  the  heavy  bars. 


In  General  Physics 


193 


It  may  be  shown  that  in  this  form  of  the  Wheatstone 
bridge  the  calculated  result  is  least  in  error  when  X = R 
and  P = Q.  It  is  therefore  desirable  to  adjust  R until  the 
balance  position  of  S (where  there  is  no  deflection  of  the 
galvanometer  when  both  K and  S are  closed)  is  as  near 
the  middle  of  the  bridge  wire  as  is  possible  with  the  avail- 
able known  resistances. 

\ When  several  resistances  rh  r2j  r3,  etc.,  are  connected 
end  to  end  in  such  a way  that  the  current  may  pass  suc- 
cessively from  one  into  the  other,  they  are  said  to  be  in 
series.  Let  the  potential  drop  in  ri  be  vh  that  in  r2  be  v2, 


© 


a 


— sAAAAA/W^^AA/ — Q 


— WM /WWW — 


id 


a 6 <tr/cs 


I0>‘ 


C»  -^VW\AAA/WW— 


a 

Q- 

A 


— /WWWWW\r~ 


~ /W\MAAAA/W— 

— wvwwvww— 


D 

D 

D 


OJ 


B 


b .Parra//c/. 


Fig.  49 


etc.,  and  let  the  total  drop  through  all  of  them  be  V . Then 
V = vi +^2+^3+  etc.,  and  since  the  current  strength  / is 
the  same  throughout  the  circuit,  we  have  on  dividing  through 
by  / 


V * 
1~~I 


v2 

1 


etc., 


but  by  Ohm’s  law  V/I  = R , the  total  resistance,  vi/I  = rh 
etc.  Therefore  this  equation  may  be  written 


R = fi+f2+r3+etc. 

or  in  words,  the  total  resistance  of  several  conductors  joined 
in  series  is  equal  to  the  sum  of  their  separate  resistances. 
This  holds  for  any  number  of  resistances  in  series.  In  figure 


194 


Manual  of  Experiments 


49a,  if  the  current  enters  by  the  terminal  A and  leaves  by 
the  terminal  B , it  must  pass  through  the  four  coils  in  series. 

If  several  resistances  are  connected  so  that  they  form  a 
divided  circuit,  by  connecting  one  end  of  each  together 
and  to  one  terminal  of  the  source  of  current,  the  other 
ends  all  being  connected  together  and  to  the  other  terminal, 
the  current  will  pass  simultaneously  through  all  of  them, 
and  they  are  said  to  be  in  parallel , (or  in  multiple  arc.) 
In  figure  49£,  if  the  current  enters  by  the  terminal  A and 
leaves  by  the  terminal  B , it  will  pass  through  the  four 
coils  in  parallel;  that  is,  it  will  divide  and  part  of  it  will 
pass  through  each  of  the  four  branches.  Those  termi- 
nals of  the  resistances  which  are  connected  together  must 
be  at  the  same  potential,  or  in  other  words,  the  drop  of  po- 
tential V in  any  one  resistance  must  equal  that  in  any  other 
resistance  and  also  that  in  their  combined  resistance  R. 
Let  the  current  strengths  in  resistances  rh  r2,  r3,  etc.,  be 
ih  i2,  is,  etc.,  respectively.  The  total  current,  I,  will  divide 
among  the  various  branches  so  that 

I — i iT_f2~f_23T-etc. 
or  by  Ohm’s  law,  since 

V V V 

I — — .?  i\  — — . 72~ — • etc. 

R ri  r2 

V V V V 

— .= 1 1 (-  etc. 

R T\  r2  7*3 

Dividing  through  by  V gives 

1111 

— = 1 1 \-  etc. 

R Ti  r2  r 3 


In  General  Physics 


195 


or  the  reciprocal  of  the  total  resistance  equals  the  sum  of  the 
reciprocals  of  the  separate  resistances.  The  reciprocal  of  the 
resistance  of  a conductor  is  called  its  conductance,  and  the 
law  may  be  stated  as  follows: — When  conductors  are  joined 
in  parallel  their  joint  conductance  is  the  sum  of  their  separate 
conductances.  Conductance  is  expressed  in  terms  of  a 
unit  called  the  mho  (the  reverse  of  ohm.)  From  the  equation 
given  it  is  seen  that  for  two  conductors  in  parallel,  the  last 
equation  may  be  written  in  the  form 


H r2  r i r2  rs 

R = , for  three,  R = etc. 

r±+r2  rir2+r2rz+r3ri 


It  should  be  noted  that  the  total  resistance  of  a number  of 
conductors  in  parallel  is  always  less  than  the  resistance 
of  any  one  of  them  alone.  ^ 

Shunts — Since  for  parallel  circuits  the  drop  of  potential 
is  the  same  through  all  of  the  branches,  in  the  case  of  two 
branches 


Therefore 


V = IR  — iiri  = i2r2 
ii  : H = r2  : n 


or  in  words,  the  currents  in  branched  circuits  are  inversely 

proportional  to  the  resistance  of  the  branches,  so  that  the 

* 

larger  part  of  the  total  current  flows  through  the  smaller 
resistance  and  vice  versa. 

From  the  equations  obtained  for  R and  V we  derive 
the  following  expressions  for  ii  and  i2: 

11  = IR/ n = Irxr2/  (n + r2)  n = Ir2/  (n + r2) 

12  = IR/ r2  = Irxr2/  ( rx+r2)r2  = In/  (n+r2) 


. Therefore 


ii  : I = r2  : (ri+r2) 


196 


Manual  of  Experiments 


and 

k : I = n : Oi +r2) 

or  in  words,  the  current  in  one  branch  of  a parallel  circuit 
is  to  the  total  current  as  the  resistance  of  the  others  branch 
is  to  the  sum  of  their  resistances. 

The  sudden  rush  of  a strong  current  through  a galvano- 
meter causes  a sudden  large  deflection  of  the  movable  sys- 
tem, which  is  likely  to  break  the  suspension.  A large  cur- 
rent may  also  overheat  the  coil  and  injure  the  insulation  of 
the  wire,  etc.  To  avoid  these  effects  a low  resistance  is 
joined  in  parallel  with  the  galvanometer,  so  that  the  larger 
part  of  the  current  is  diverted  from  it.  Such  a low  resistance 
is  called  a shunt,  and  sensitive  galvanometers  are  frequently 
supplied  with  three  shunts  of  such  resistance  that  only 
0.1,  0.01,  or  0.001  of  the  total  current  may  pass  through  the 
galvanometer  when  they  are  joined  in  parallel  with  it. 
If  S be  the  resistance  of  the  shunt,  G that  of  the  galvano- 
meter, / the  total  current  and  Ig  that  through  the  galvano- 
meter, we  see  from  the  above  theory  of  parallel  circuits  that 

Ig  \ I — S \ S-\-G 

or 

S 

Ig  = I— 

S + G • 

so  that  for  values  of  S that  are  small  compared  with  G, 
the  galvanometer  current  is  a small  fraction  of  the  total 
current. 

The  shunt  box  shown  in  Figure  50,  is  connected  to  the 
galvanometer  by  means  of  the  binding  posts  A and  B.  Any 
one  of  the  three  shunt  coils  may  be  placed  in  parallel  with 
the  galvanometer  coil  by  putting  the  plug  in  the  gap  between 
that  coil  and  the  block  of  metal  to  which  the  binding  post 
B is  connected. 


In  General  Physics 


197 


Directions. — Connect  the  apparatus  as  shown  in  the 
diagram  (Fig.  48).  The  unknown  resistance  consists  of 
four  coils,  each  having  its  ends  joined  to  binding  posts  at 
opposite  sides  of  the  box.  (See  Fig.  49).  Connect  the  coil 
marked  No.  1 with  the  bridge  and, 'with  the  smallest  value 


of  R in  circuit  and  the  key  S at  the  middle  of  the 
bridge  wire,  note  the  deflection  of  the  galvanometer 
when  both  K and  S are  closed.  K should  always  be  closed 
first,  and  since  an  open  circuit  cell  is  used  it  should  not  be 
kept  closed  longer  than  necessary.  S should  never  be  pressed 
down  on  the  wire  while  it  is  being  moved.  The  same  pres- 
sure should  be  used  on  the  keys  at  all  times. 

Increase  R and  note  whether  the  direction  of  the  deflec- 
tion is  opposite  to  what  it  was  before.  If  not,  try  a still 
greater  value  of  R.  In  this  way  find  two  values  of  R for 
which  the  deflections  are  in  opposite  directions.  The  value 


198 


Manual  of  Experiments 


of  X must  evidently  lie  somewhere  between  these  two 
values  of  R , and  when  the  deflection  is  in  one  direction  R 
is  too.  small,  when  in  the  opposite  direction  R is  too  large. 
With  these  relations  fixed  in  mind,  vary  R by  smaller  and 
smaller  steps  until  the  minimum  deflection  is  reached, 
then  shift  the  contact  key  S to  one  side  or  the  other  in  steps 
of  a half  millimeter  or  less,  until  there  is  no  further  decrease 
in  the  deflection  of  the  galvanometer  when  both  K and  S 
are  closed.  S should  be  kept  as  near  the  middle  of  the 
bridge  wire  as  possible.  During  all  of  the  preliminary  trials 
a low  value  shunt  should  be  connected  across  the  terminals 
of  the  galvanometer  but  when  the  balance  position  has  been 
approximately  determined  the  shunt  may  be  removed,  in 
order  to  make  the  final  adjustment.  The  balance  position 
may  also  be  determined  by  sliding  S from  one  end  of  the 
bridge  toward  the  center  until  equilibrium  is  obtained, 
then  from  the  other  end  of  the  bridge  toward  the  center 
until  no  deflection  is  obtained.  If  the  two  balance  positions 
do  not  coincide,  their  mean  should  be  taken  as  the  balance 
position.  As  a check  X and  R may  be  interchanged  and 
balance  again  obtained.  Of  course  x and  l — x should  inter- 
change their  values  also.  Note  the  value  of  R , x and  / 
and  repeat  the  performance  with  each  of  the  other  three 
coils.  Then  do  the  same  with  all  the  coils  in  series,  with 
1 and  2 in  parallel,  and  finally  with  all  of  them  in  parallel. 
Calculate  the  resistance  and  conductance  in  each  case, 
and  show  that  the  laws  for  parallel  and  series  connection 
of  resistances  hold. 

Questions. — 1. — From  the  general  expression  for  the 
total  resistance  of  a number  of  resistances  connected  in 
parallel,  show  that  the  total  resistance  of  n coils  in  parallel, 
each  of  resistance  r will  be  r/n. 

2. — What  resistance  would  be  required  in  series  with 


In  General  Physics 


199 


a galvanometer  whose  resistance  is  <2(  = 200)  ohms,  in  a 
circuit  with  an  electromotive  force  of  b(  = l)  volt,  in  order 
that  the  current  through  the  galvanometer  may  be  c(  = 
0.0001)  ampere? 

3.  — If  in  place  of  the  resistance  calculated  in  question  2 
we  put  <7(  = 160)  ohms  in  series  with  the  galvanometer? 
what  resistance  used  as  a shunt  to  the  galvanometer,  would 
give  the  same  galvanometer  current  with  the  same  electro- 
motive force?  ~ 

4.  — a)  Given  three  resistance  coils  A,  B and  C,  with 
resistances  a(  = 1),  b(  = 2)  and  c(  = 3)  ohms  respectively, 
describe  (or  sketch)  the  eight  possible  combinations  of  all 
three  coils  (arranged  in  series,  or  parallel,  or  combinations 
of  the  two)which  will  give  a different  total  resistance. 

b)  Calculate  the  total  resistance  in  each  case. 

37.  THE  POST  OFFICE  BOX  BRIDGE 

To  determine  the  specific  resistance  of  several  wires  and 
the  variation  of  their  resistance  with  length  and  cross 
section  (D.  442-447,  456;  G.  923,  844,  845,  850,  856;  K.  641, 
642,  646-649,  651;  W.  480-484,  486,  487.) 

Apparatus. — Post  office  box  bridge,  dry  cell,  D ’Arsonval 
galvanometer,  board  with  unknown  resistances  of  man- 
ganin  and  German  silver,  six  connecting  wires. 

Theory  and  Description. — The  resistance  of  a con- 
ductor may  be  defined  as  the  ratio  of  the  e.  m.  f.  applied  to  its 
ends  and  the  resulting  current  strength  flowing  through  it.  It 
may  be  shown  that  the  resistance  of  a conductor  is  directly 
proportional  to  its  length  /,  and  inversely  proportional  to 
its  cross  section  A,  and  that  it  depends  upon  the  material 
of  the  conductor,  or  in  symbols 

R = p {1/ A) 


200 


Manual  of  Experiments 


The  factor  p is  a constant  for  any  one  substance  at  a given 
temperature,  and  is  called  the  specific  resistance  of  the  sub- 
stance, or  its  resistivity.  If  both  / and  A are  unity,  R will 
equal  />,  hence  the  specific  resistance  of  a substance  may  be  de- 
fined as  the  resistance  of  a conductor  of  that  substance  having  a 
length  of  one  centimeter  and  a cross  section  of  one  square  cen- 
timeter. 

The  Post  Office  box  is  a compact  form  of  the  Wheatstone 
bridge,  in  which  there  are  two  identical  sets  of  resistance 
coils  ( A to  G in  Fig.  51)  generally  of  10,  100  and  1000  ohms', 
called  the  ratio  arms  of  the  bridge,  and  the  usual  set  of 
known  resistance  coils  R,  (between  A and  X ) with  two 
contact  keys,  all  contained  in  the  same  box.  The  con- 


In  General  Physics 


201 


nections  are  somewhat  different  from  those  of  the  slide 
wire  bridge.  There  is  a fixed  connection  from  the  junction 
of  the  two  ratio  arms  to  the  left  hand  key,  and  another 
from  the  block  A to  the  right  hand  key,  by  means  of  wires 
within  the  box.  The  galvanic  cell  is  to  be  connected  to 
the  binding  posts  B and  X,  the  galvanometer  to  the  bind- 
ing posts  G and  Gi,  and  the  unknown  resistance  X between 
the  binding  posts  marked  X and  G. 

If  we  let  R\  be  the  ratio  arm  adjoining  G and  R2  the  one 
adjoining  A,  and  represent  Ri,  R2,  R and  X (in  the  order 
given)  as  the  sides  of  the  diamond  shaped  figure  which  is 
the  usual  diagrammatic  representation  of  the  Wheatstone 
bridge,  we  find  on  tracing  out  the  connections  in  Fig.  51, 
that  the  galvanic  cell  is  connected  between  the  junction 
of  the  two  ratio  arms  and  the  junction  of  the  known  and 
unknown  resistances;  and  that  the  galvanometer  is  connected 
across  the  two  ratio  arms.  It  is  readily  shown  by  applying 
the  same  train  of  reasoning  to  this  diagram  as  was  used  in 
the  case  of  the  slide  wire  bridge,  that  for  values  of  R , Rh 
and  R2  which  balance  the  bridge  (give  zero  deflection  of 
the  galvanometer)  the  unknown  resistance  may  be  cal- 
culated from  the  proportion 

X : R = Ri  : R2 

From  this  proportion  it  may  be  seen  that  if  X be  greater 
than  R , then  Ri  must  be  greater  than  R2 , and  vice  versa. 

Since  X = RxRi/ R2  the  smallest  unknown  resistance 
which  can  be  measured  with  this  bridge  is  the  smallest 
fraction  R\/R2  (which  can  be  formed  with  the  fixed  values 
of  R\  and  R2)  of  the  smallest  of  the  known  resistances  R. 
In  other  words  if  Ri  and  R2  are  set  at  10  and  1000  ohms  re- 
spectively, and  the  smallest  of  the  known  resistance  coil 
is  1 ohm,  the  unknown  resistance  can  be  determined  to 


202 


Manual  of  Experiments 


1/100  of  an  ohm,  whereas  if  Ri  and  i?2  are  set  at  10  and  100 
ohms  respectively,  the  unknown  resistance  may  be  deter- 
mined to  only  1/10  ohm. 

In  any  form  of  Wheatstone  bridge  there  are  six  re- 
sistances to  be  considered,  namely  the  known  and  un- 
known resistances,  that  of  the  galvanometer,  that  of  the 
cell,  and  those  of  the  ratio  arms  (or  of  the  two  sections  of 
wire  in  the  slide  wire  bridge.)  In  books  on  advanced 
Physics  it  is  shown  that  the  most  sensitive  arrangement  of 
the  bridge  is  to  have  the  greater  of  the  two  resistances, 
that  of  the  galvanometer  and  that  of  the  galvanic  cell, 
connected  between  the  junction  of  the  two  larger  remaining 
resistances  and  the  junction  of  the  two  smaller  ones.  In 
general  the  galvanometer  has  a greater  resistance  than  the 
cell,  and  since  J and  are  either  greater  or  less  than  R 
and  i?2  respectively  the  arrangement  of  the  bridge  shown 
in  fig.  51  is  the  most  sensitive  one. 

Directions. — Connect  the  apparatus  as  shown  in  the 
diagram.  The  unknown  resistance  consists  of  four  wires. 
Three  of  these  are  of  manganin,  two  of  which  have  the  same 
length  but  different  diameters,  while  two  others  have  the 
same  diameter  but  different  lengths.  The  fourth  wire  is 
of  German  Silver.  The  resistance  of  each  is  to  be  de- 
termined. 

Having  connected  one  of  the  unknown  resistances  to 
the  blocks  X and  G,  find  the  approximate  resistance  of  X 
by  making  the  ratio  arms  equal  (say  100  ohms  each)  and 
varying  R until  the  galvanometer  deflection  reaches  its 
minimum  value.  Note  that  if  R be  started  at  a small 
value  and  gradually  increased,  the  galvanometer  deflections 
will  decrease  to  zero  when  the  bridge  is  balanced.  If  R 
be  still  further  increased  the  galvanometer  will  be  deflected 
in  the  opposite  direction  from  what  it  was  at  first.  De- 


In  General  Physics 


203 


flections  of  the  galvanometer  in  the  first  direction  then 
indicate  that  R is  too  small  for  balance,  whereas  deflections 
in  the  opposite  direction  correspond  to  values  of  R too  large 
for  balance.  These  relations  should  be  borne  in  mind 
throughout  the  experiment.  On  account  of  the  fixed  re- 
sistance of  the  coils  in  R it  may  not  be  possible  to  obtain 
perfect  balance  (zero  deflection),  but  when  the  deflection 
of  the  galvanometer  is  a minimum.,  R is  approximately 
equal  to  X.  Now  find  the  approximate  ratio  between  the 
approximate  value  of  X and  the  total  available  resistance 
of  R and  adjust  the  ratio  arms,  R\  and  R2,  to  approximately 
the  same  ratio.  Vary  R until  balance  is  obtained  (minimum 
deflection  of  galvanometer),  when  both  keys  are  closed. 
Note  the  values  of  R , R\  and  R2,  and  calculate  X.  In 
the  same  way  determine  each  of  the  unknown  resistances. 

Note  the  length,  diameter  and  material  of  each  wire  and 
calculate  the  resistance  per  centimeter,  and  the  area  of  cross 
section . 

To  find  the  relation  between  the  resistance  of  a wire  of 
any  given  material  and  one  of  its  dimensions,  all  of  the  other 
dimensions  must  be  kept  constant.  Keeping  this  in  mind 
show  with  your  data  that  the  resistance  varies  directly  as 
the  length  and  inversely  as  the  cross  section  of  the  conductor 
by  calculating  the  ratios  of  X and  L for  wires  of  the  same 
material  and  diameter,  and  the  products  of  X and  A for 
wires  of  the  same  material  and  length.  Also  calculate  the 
specific  resistance  of  manganin  and  of  German  silver. 

Questions. — 1. — In  Electrical  Engineering  the  ex- 
pression “resistance  per  mil-foot”  is  often  used,  meaning 
the  resistance  of  a conductor  one  foot  long  having  a cross 
sectional  area  equal  to  that  of  a circle  whose  diameter  is 
one  mil  (one  thousandth  inch).  Find  the  ratio  between 
specific  resistance  and  resistance  per  mil-foot  of  a substance. 


204 


Manual  of  Experiments 


2.  — If  a cylindrical  conductor  were  stretched  to  a(  = 2) 
times  its  original  length,  assuming  its  volume  to  remain 
constant,  how  would  its  resistance  compare  with  its  original 
resistance? 

3.  — An  all  metal  telephone  line  (two  wires)  has  a re- 
sistance of  a(  = 2A0)  ohms  when  measured  at  the  station, 
indicating  that  it  is  short  circuited  somewhere.  If  the 
wire  is  of  copper  with  cross  section  £(  = 0.1)  sq.  cm.  and 
specific  resistance  1.5X10-5  ohms  per  c.  c.  what  is  the 
distance  of  the  short  circuit  from  the  station? 

4.  — In  measuring  an  unknown  resistance  with  a P.  0. 
box  bridge,  the  ratio  arms  were  set  at  a(  = 1000)  and  £(  = 10) 
ohms  respectively,  and  balance  was  obtained  when  the 
adjustable  resistance  (connected  to  the  smaller  ratio  arm) 
was  c{  = 750)  ohms.  The  P.D.  of  the  cell  was  ^(  = 1.4)  volts. 
Compute  a)  the  unknown  resistance,  b)  the  current  through 
it,  c ) the  P.  D.  across  it,  and  d)  the  current  through  the  cell. 

38.  THE  POTENTIOMETER. 

To  determine  the  electromotive  force  of  several  galvanic 
cells,  singly  and  in  combination,  by  means  of  the  poten- 
tiometer. (D.  441,  454,  455  469-476;  G.  812-831,  844-849, 
925,  927;  K.  622-642;  W.  480,  490,  550,  557.) 

Apparatus. — Pye  potentiometer  with  two  storage  cells, 
D’Arsonval  galvanometer,  dry  cell,  salammoniac  cell,  Daniel 
cell,  two  gravity  cells  and  eight  connecting  wires. 

Description  and  Theory. — A galvanic  cell  consists  in 
general  of  two  plates  of  different  metals  (or  a plate  each  of 
a metal  and  some  other  substance  such  as  carbon)  placed  a 
short  distance  apart  in  an  acid  or  a solution  of  a salt,  such 
that  a difference  of  potential  exists  between  the  plates,  and 
a current  of  electricity  will  flow  from  one  to  the  other  when 


In  General  Physics 


205 


their  external  ends  are  joined  by  means  of  a conductor. 
The  electromotive  force  of  such  a cell  is  equal  to  the  difference 
of  potential  existing  between  the  external  ends  of  these  plates 
when  they  are  not  connected , (i.  e.  there  is  an  open  circuit) 
or  practically  when  they  are  joined  by  a very  high  resistance. 

The  electromotive  force  of  cells  may  be  compared  by 
the  potentiometer  method.  The  simple  potentiometer  con- 
sists of  a long  homogeneous  wire  AD  (Fig.  52)  of  uniform 
cross  section.  A constant  fall  of  potential  is  produced  in 
this  wire  by  means  of  a source  of  constant  current  (in  this 
case  a storage  battery  S.  B.)  By  Ohm’s  law  the  potential 
drop  through  a wire  is  equal  to  the  current  through  it  times 
its  resistance.  Since  our  wire  is  uniform  and  homogeneous, 
its  resistance  per  unit  length  is  the  same  throughout  its 
length,  and  since  the  electromotive  force  ( e.m.f .)  of  the  stor- 
age cells  is  practically  constant,  the  current  strength  is 
constant.  It  follows  that  the  potential  drop  per  unit  length 
of  wire  is  the  same  at  all  points. 

A galvanic  cell  B is  now  connected  in  series  with  a sen- 
sitive galvanometer  G to  one  end  A of  the  wire  and  by 
means  of  a sliding  contact  key  S to  another  point  on  the  wire. 
This  second  cell  also  tends  to  produce  a constant  potential 
drop  through  AS.  But  AS  is  in  parallel  (See  Exp.  36)  with 
both  of  the  circuits ' containing  cells,  and  if  like  terminals 
(either  -f-  or  — ) of  both  cells  are  connected  to  the  end  A 
their  e.m.fd s will  be  opposed.  The  current  from  cell  B 
will  tend  to  divide  at  A part  going  through  AS  and  part 
through  the  circuit  A(SB)DS.  The  current  from  S.B.  will 
tend  to  divide  at  the  same  point,  part  going  through  AS  and 
part  through  the  circuit  ABGKS.  Each  branch  current  will 
be  opposed  by  the  electromotive  force  of  the  other  cell  and 
if  the  drop  of  potential  through  AS  is  not  the  same  for  both 
cells,  a current  will  flow  through  it  in  one  direction  or  the 


206 


Manual  of  Experiments 


other,  depending  upon  which  cell  produces  the  greater  drop, 
and  there  will  be  in  general  a deflection  of  the  galvanometer. 
If  now  the  sliding  contact  key  S be  moved  along  the  wire 
a point  may  be  found  where  there  is  no  deflection  of  the 


Fig.  52 


galvanometer  when  the  key  is  closed,  because  there  is  no 
current  flowing  in  the  galvanometer  circuit.  This  indicates 
that  the  potential  drop  between  A and  S due  to  cell  B is 
exactly  equal  to  that  due  to  cells  S.B. , or  in  other  words  the 
tendency  for  cell  B to  send  current  through  the  galvano- 
meter circuit  from  S to  A is  exactly  neutralized  by  the  ten- 
dency of  cells  S.B.  to  send  current  through  the  same  circuit. 
Since  there  is  no  current  through  the  galvanometer  circuit 
there  will  be  no  potential  drop  between  the  cell  B and  the 
points  A and  S so  that  the  point  A must  be  at  the  same 
potential  as  the  positive  terminal  of  B and  the  point  S 
at  the  same  potential  as  the  negative  terminal.  Con- 
sequently the  e.m.f.  of  the  cell  B must  be  equal  to  the  po- 
tential drop  in  the  section  of  wire  AS  due  to  the  cells  S.B., 
and  is  therefore  directly  proportional  to  the  length  of  wire 
AS. 

A second  cell  C of  known  e.  m.  /.,  such  as  a standard  or 
a Daniel  cell,  may  now  be  inserted  in  place  of  B and  the 
contact  key  moved  along  the  wire  to  some  point  S'  where 
there  will  be  no  deflection  of  the  galvanometer  when  the 
key  is  closed.  The  e.m.f.  of  cell  C is  then  directly  propor- 
tional to  the  length  of  wire  AS'. 


In  General  Physics 


207 


If  E be  the  e.m.f.  of  a cell,  L the  length  and  R the  re- 
sistance of  the  stretch  of  wire  across  which  it  is  balanced, 
P.  D.  the  potential  drop  through  it,  and  I the  current 
flowing  in  it,  each  with  their  proper  subscript,  we  may 
write 


Therefore 


E b P.D.  b IRb  Lb 


Ec  P.D.  c IRc  Lc 


Lb 

FjB  = Ec — — 

Lc 


The  Pye  potentiometer  has  a uniform  high  resistance 
wire,  ten  meters  long,  wound  back  and  forth  for  convenience 
sake,  in  ten  lengths  of  one  meter  each,  on  a board  supplied 
with  a millimeter  scale.  (See  Fig.  53).  The  scale  is 
calibrated  from  both  ends  so  that  lengths  of  wire  from  either 
end  of  a loop  may  be  measured  directly.  The  key  S,  is 
carried  by  a sliding  triangular  bridge,  so  that  contact  may 
be  made  at  any  point  on  any  one  of  the  ten  lengths  of  wire. 
The  storage  batteries  are  connected  to  terminals  A and  D of 
the  potentiometer  wire,  the  primary  cell  circuit  to  A and  to 
E or  F,  like  poles  of  both  being  connected  to  A.  Terminals 
E and  F are  at  the  ends  of  a metal  strip  on  which  one  leg 
of  the  bridge  rests,  so  that  the  current  may  pass  from  E or 
F to  S through  the  bridge.  The  other  two  legs  of  the  bridge 
rest  on  the  wooden  base  and  are  therefore  insulated.  Care 
must  be  taken  that  the  bridge  does  not  come  in  contact 
with  the  binding  post  A,  as  this  will  cause  a shortcircuit. 
When  the  potentiometer  is  balanced,  the  length  of  wire  from 
binding  post  A to  key  S may  be  read  on  the  attached  scale. 

The  Daniel  cell  has  a sheet  of  copper  bent  into  cylin- 
drical shape  and  immersed  in  a copper  sulphate  solution. 
A small  pocket  on  the  plate  contains  crystals  of  blue  vitriol 


208 


Manual  of  Experiments 


for  keeping  up  the  concentration  of  the  solution.  Within 
this  cylinder  is  placed  a porous  cup  containing  a zinc  rod  in 
dilute  sulphuric  acid.  When  properly  set  up  this  cell  has 
an  e.m.f.  of  1.08  volts,  which  with  proper  care  remains  con- 
stant for  a long  time.  The  Daniel  cell  may  therefore  be 
used  as  a standard  for  comparison. 

The  gravity  cell  has  the  same  elements  as  the  Daniel 
cell,  without  the  porous  cup,  depending  upon  the  difference 
in  the  specific  gravities  of  the  copper  sulphate  and  the  zinc 
sulphate  formed  to  keep  the  liquids  apart. 

The  salammoniac  cell  has  carbon  and  zinc  plates  in  a 
solution  of  ammonium  chloride,  and  the  dry  cell  is  practically 
the  same  except  that  the  carbon  plate  is  packed  in  some 
material  that  absorbs  the  solution  and  prevents  polariza- 
tion, and  is  contained  in  a zinc  cylinder. 


Fig.  53 


The  storage  cells,  in  their  common  form,  consist  of  two 
lead  grids  having  their  interstices  filled  with  lead  sulphate 
formed  by  making  a paste  of  red  lead  and  dilute  sulphuric 
acid.  These  plates  are  immersed  in  a weak  solution  of  sul- 
phuric acid  and  are  charged  by  passing  a current  from  one 
to  the  other.  On  connecting  the  two  together  externally  a 
current  will  flow  in  the  opposite  direction  to  that  of  the 
charging  current.  They  are  charged  and  discharged  in 
this  way  a number  of  times  to  form  them. 

The  Daniel  and  gravity  cells,  and  the  storage  cells  may 
be  left  in  a closed  circuit  for  some  time  without  their  e.m.f. 
depreciating  markedly,  i.  e.  they  are  closed  circuit  cells  and 


In  General  Physics 


209 


do  not  polarize  easily.  The  salammoniac  and  dry  cells  are 
however  open  circuit  cells,  and  can  be  used  only  for  short 
intervals  at  a time,  because  their  e.m.f.  drops  when  they 
are  in  use  due  to  polarization,  and  only  recovers  when  they 
have  been  left  on  open  circuit  for  some  time. 

In  order  that  we  may  have  the  proper  current  strength 
or  electromotive  force  in  a given  circuit,  it  is  sometimes 
necessary  to  connect  several  cells  in  series,  or  in  parallel, 
or  in  combinations  of  the  two.  (See  Fig.  54.)  To  connect 
them  in  series  the  positive  terminal  of  one  is  joined  to  the 
negative  terminal  of  the  next,  etc.,  leaving  one  free  positive 
and  one  free  negative  terminal  to  be  connected  to  the  given 
circuit.  The  potential  difference  from  one  end  of  the 
series  to  the  other  is  evidently  the  sum  of  the  potential 
drops  through  each  cell,  so  that  the  combined  e.m.f . of  the 
cells  in  series  is  the  sum  of  their  separate  e.m.f ’s.  But 
it  must  be  remembered  that  the  effective  (internal)  resistance 
of  such  a cell  is  that  of  the  liquid  through  which  the  current 
passes  from  one  plate  to  the  other  (the  resistance  of  the  plates 


Fig.  54 


themselves  being  negligible),  and  when  several  cells  are  in 
series  the  current  must  pass  through  their  respective  liquids 
successively,  or  the  total  resistance  of  the  cells  in  series  is 
the  sum  of  their  separate  resistances.  The*  current  strength 
may  be  determined  by  Ohm’s  law,  the  total  resistance  in- 
cluding the  internal  resistance  of  the  cell  and  the  external 
resistance  of  the  circuit,  so  that  on  short  circuit  (zero  external 
resistance)  the  current  strength  is  the  same  as  that  in  a single 
cell.  When  several  cells  are  connected  in  parallel,  all  the 
positive  terminals  are  joined  together  and  to  one  end  of 


210 


Manual  of  Experiments 


the  circuit,  and  all  the  negative  terminals  together  and  to 
the  other  end  of  the  circuit.  The  potential  difference  from 
one  set  of  terminals  to  the  other  is  evidently  the  same  as 
the  potential  drop  through  a single  cell  if  they  are  all  of 
the  same  kind,  and  the  combined  e.m.f.  of  the  cells  in  parallel 
is  the  e.m.f.  of  a single  cell.  But  the  current  passes  through 
all  of  them  simultaneously  and  though  the  length  of  the 
liquid  path  is  the  same  for  all  of  them  as  for  one  cell,  the 
cross  section  for  n of  them  in  paralled  is  n times  as  large  as 
that  for  a single  cell,  and  since  the  resistance  of  a conductor 
varies  inversely  as  its  cross  section,  the  resistance  in  this 
case  will  be  1/n  times  that  of  a single  cell.  As  a result 
the  current  would  be  n times  as  great  as  from  a single  cell 
on  short  circuit 

It  is  sometimes  advantageous  to  combine  both  methods 
of  connecting  cells.  For  instance  if  we  have  6 cells,  we  may 
connect  them  in  2 rows,  each  row  having  3 cells  in  series. 
The  2 rows  may  then  be  joined  in  parallel  by  connecting 
their  free  positive  terminals  together  at  one  end  of  the 
circuit  and  their  free  negative  terminals  together  at  the 
other  end.  Or  we  may  connect  them  in  3 rows  each  having 
2 cells  in  series  and  then  join  these  3 rows  in  parallel. 

It  may  be  shown  that  in  order  to  obtain  the  maximum 
current  from  a combination  of  cells  in  a given  circuit,  they 
should  be  arranged  in  such  a way  that  the  internal  resistance 
of  the  cells  is  equal  to  the  external  resistance  of  the  circuit. 

In  the  general  case  let  n be  the  number  of  cells  of  the 
same  kind  to  be  connected,  s the  number  in  series  in  each 
row,  p the  number  of  rows  joined  in  parallel,  r the  resistance 
and  e the  e.m.f.  of  each  cell,  and  R the  external  resistance. 
Then  the  total  number  of  cells  will  be  n = ps.  The  e.m.f. 
of  each  row  will  be  se  and  its  resistance  sr . The  e.m.f. 
of  p rows  in  parallel  will  be  se  also,  but  their  combined 


In  General  Physics 


211 


resistance  will  be  sr/p.  The  current  in  the  external  circuit 
is  therefore 

se  pse  ne 

1 = = — = 

R-\-sr/p  pR-\-sr  pR-\-sr 

This  current  will  evidently  be  a maximum  when  the  de- 
nominator pR+sr  is  a minimum.  Suppose  pR  and  sr 
to  be  the  sides  of  a rectangle,  their  product  being  its  area. 
This  product  psRr  is  a constant  in  any  given  case,  for  the 
number  of  cells  (ps)  is  fixed,  the  external  resistance  is 
fixed,  and  for  a given  current  the  internal  resistance  of 
the  cells  will  be  constant.  But  it  is  known  that  of  all 
rectangles  of  a constant  area  that  one  has  the  least  peri- 
meter which  has  its  sides  equal,  (the  square.)  There- 
fore the  sum  of  two  sides  pR-fsr  will  be  a minimum 
when  pR  = sr , or  when  R=sr/p.  In  other  words,  in  a 
series-parallel  combination  of  cells , the  current  will  have 
its  maximum  value  when  the  external  and  total  internal  re- 
sistance are  equal. 

When  two  cells  have  two  like  terminals  joined  to- 
gether, and  two  to  the  external  circuit,  their  e.m.f.'s  are 
opposed  to  each  other  and  their  combined  e.m.f.  is  the  dif- 
ference of  their  separate  e.m.f.'s. 

Directions. — Connect  the  apparatus  as  shown  in  the 
diagram,  being  careful  to  have  the  cells  to  be  tested  opposed 
to  the  storage  cells,  and  the  latter  in  series  with  each  other. 
The  Daniel  cell  to  be  used  as  a standard  is  to  be  tested  first. 
By  means  of  the  sliding  key,  contact  should  be  made  with 
one  wire  after  the  other  until  two  adjacent  wires  are  found  for 
wThich  the  galvanometer  gives  deflections  in  opposite  direc- 
tions. The  key  is  then  moved  along  between  the  contact 
points  on  these  two  wires  and  contact  made  at  different 
points  until  the  position  s'  is  found  for  which  there  is  no  gal- 


212  Manual  of  Experiments 

variometer  deflection.  The  length  of  wire  AS'  should  be 
noted.  The  key  must  not  be  held  in  contact  with  the  wire 
while  it  is  being  moved,  and  should  not  be  pressed  hard  at 
any  time,  as  this  may  injure  the  wire.  Each  of  the  gravity 
cells  is  then  to  be  put  in  place  of  the  standard  and  the 
balance  positions  found,  the  lengths  AS  being  noted  in  each 
case.  This  is  repeated  with  both  gravity  cells  in  parallel, 
then  in  series,  and  then  in  opposition.  In  the  latter  case  the 
cell  having  the  higher  e.m.f.  should  be  connected  so  that  it 
has  the  same  terminals  (+  or  — ) joined  to  A and  D as  the 
storage  cells  have.  Finally  the  dry  cell  and  the  salam- 
moniac  cell  are  each  connected  to  the  potentiometer  and 
balanced.  From  the  data  obtained  the  e.m.f.  of  each  cell 
and  of  each  combination  is  to  be  determined  by  comparison 
with  that  of  the  Daniel  cell.  The  e.m.f.  of  each  combination 
should  then  be  compared  with  the  separate  e.m.f. 's  of  the 
two  gravity  cells. 

Questions. — 1. — What  theoretical  arrangement  of  <z(  = 20) 
cells,  each  having  an  e.m.f.  of  &(  = 1.9)  volt,  and  an  internal 
resistance  of  c(  = 0.2)  ohm,  will  send  the  maximum  current 
through  an  external  resistance  of  ^(  = 0.1975)  ohm?  Find 
the  theoretical  maximum  current  strength. 

2.  — Find  the  current  strength  through  the  above  ex- 
ternal resistance  from  these  cells  when  arranged  in  all 
possible  series-parallel  combinations;  that  is,  in  parallel 
rows,  each  row  having  1,  2,  4,  5,  10  or  20  cells  in  series 
respectively.  What  is  the  actual  maximum  current  obtain- 
able? 

3.  — Plot  a curve  with  the  data  obtained  in  question,  2 
having  the  number  of  cells  in  series  as  abscissae  and  the 
corresponding  current  strength  as  ordinates. 

4.  — A certain  galvanic  cell  will  send  a current  of  #(  = 0.28) 
ampere  through  an  external  resistance  of  b(  = 3.)  ohms. 


In  General  Physics 


213 


When  this  resistance  is  changed  to  c(  = 8.)ohms  the  current 
drops  to  ^(  = 0.14)  amperes.  Calculate  thee.m.f.  and  internal 
resistance  of  the  cell. 

39.  FIGURE  OF  MERIT  OF  A GALVANOMETER. 

To  determine  the  figure  of  merit  of  a D’Arsonval  gal- 
vanometer. (D.  437,  438,  452;  G.  854;  K.  602-605,  645;  W. 
477,  480,  486.) 

Apparatus. — Resistance  box  (10,000  ohms)  traveling 
plug,  reversing  switch,  D’Arsonval  galvanometer,  galvanic 
cell,  and  connecting  wires. 

Theory  and  Method. — The  figure  of  merit  of  a galvano- 
meter is  defined  as  the  strength  of  that  current  which  on  passing 
through  the  coil  of  the  galvanometer  produces  a deflection  of 
one  scale  division . If  the  deflections  are  observed  by  the 
telescope  and  scale  method,  the  figure  of  merit  is  usually 
calculated  for  a deflection  of  one  millimeter  with  a distance 
of  one  meter  between  scale  and  mirror.  The  method  is  an 
application  of  Ohm’s  law.  The  apparatus  is  arranged  as 
shown  in  the  diagram,  Fig.  55.  A gravity  cell  of  known 
e.m.f.  is  connected  in  series  with  the  D’Arsonval  galvano- 
meter G of  known  resistance,  and  a variable  resistance  R 
through  a reversing  switch.  The  galvanometer  coil  of  fine 


wire  cannot  safely  stand  a current  of  more  than  about 
0.00001  ampere,  and  with  an  e.m.f.  of  one  volt  in  the  circuit, 
R would  have  to  be  very  large  to  reduce  the  current  strength 
to  this  amount.  On  the  other  hand  if  the  current  in  the  gal- 


214 


Manual  op  Experiments 


? 


vanometer  were  reduced  by  using  a shunt,  (the  resistance 
R being  omitted)  the  shunt  resistance  would  have  to  be  in- 
conveniently small.  A combination  of  both  series  and  shunt 
resistances  is  therefore  used,  -as  shown  in  the  diagram. 

Let  S be  the  resistance  of  the  shunt,  G that  of  the  gal- 
vanometer, B that  of  the  cell,  and  A the  e.m.f.  of  the  latter. 
The  total  resistance  of  the  circuit  is  then  R+B+(SG/ (S+G)) 
and  the  current  strength  in  the  main  circuit  is  by  Ohm’s 
law, 

E 

/ = — 


R+B+ 


SG 

S+G 


The  resistance  of  the  cell  and  that  of  the  galvanometer 
and  shunt  in  parallel  are  so  small  compared  with  the  values 
of  R to  be  used,  that  they  may  be  neglected  in  calculating 
/.  But  the  current  in  the  galvanometer  is 


S 

h = I 

S+G 


(See  Exp.  36)  and  if  d be  the  deflection  in  millimeters  cor- 
responding to  this  current  strength,  the  figure  of  merit  will  be 

h IS  ES 

E = = = 

d d(S+G)  Rd{S+G) 

A galvanometer  may  be  calibrated  to  be  used  either  as 
an  ammeter,  an  instrument  for  measuring  current  strength 
in  amperes,  or  as  a voltmeter,  an  instrument  for  measuring 
the  drop  in  potential  through  a circuit,  in  volts. 

Ammeters. — As  an  ammeter  it  must  be  connected  in 
series  with  the  apparatus  through  which  the  current  is 
flowing  and  should  not  appreciably  alter  the  total  resistance 
of  the  circuit,  as  this  would  change  the  current  to  be  measured. 


In  General  Physics 


215 


Since  the  galvanometer  usually  has  considerable  resist- 
ance, it  would  appreciably  reduce  the  current  strength 
in  the  circuit  when  it  is  introduced.  On  the  other  hand  the 
galvanometer  must  be  protected  from  heavy  currents  by 
means  of  a shunts,  and  as  their  combined  resistance  SG/ 
(S  + G)  is  less  than  that  of  either  S or  G alone,  we  may,  by 
using  a suitably  small  shunt,  make  the  combined  resistance 
as  small  as  we  please.  Hence  the  change  in  current  strength 
when  the  galvanometer  and  shunt  are  inserted  in  the  circuit, 
may  be  made  negligible.  The  galvanometer  and  shunt 
resistance  may  be  so  chosen  for  a given  circuit,  that  the 
deflections  per  unit  current  will  have  any  value  we  please, 
say  one  centimeter  per  ampere  of  current  in  the  main  circuit. 

Voltmeters. — When  used  to  measure  the  drop  in 
potential  through  any  part  of  a circuit,  the  galvanometer 
must  be  placed  in  parallel  with  it,  so  that  there  is  the  same 
drop  of  potential  through  both.  At  the  same  time  the 
current  strength  through  the  circuit  should  not  be  appreci- 
ably altered,  for  this  would  alter  the  drop  of  potential  which 
is  to  be  measured,  the  two  being  directly  proportional, 
according  to  Ohm’s  law. 

If  R be  the -resistance  of  the  circuit  across  which  the 
potential'  difference  is  to  be  measured,  and  G that  of  the 
galvanometer,  their  combined  resistance  in  parallel  is 
RG/(R+G).  The  greater  the  value  of  G the  smaller  will 
be  the  difference  between  R and  RG/(R+G),  for  G/(R+G) 
will  be  nearly  unity.  If  then,  the  resistance  of  the  gal- 
vanometer circuit  be  very  large,  its  insertion  in  parallel 
with  the  main  circuit  will  not  appreciably  alter  the  total 
resistance  of  the  latter,  nor  the  current  strength,  hence  the 
drop  of  potential  through  the  circuit  will  be  practically  the 
same  before  and  after  inserting  the  galvanometer.  If  the 
galvanometer  itself  has  not  a high  resistance,  a high  re- 


216 


Manual  of  Experiments 


sistance  may  be  placed  in  series  with  it.  The  current 
strength  through  the  galvanometer  (and  therefore  its  de- 
flections also)  will  be  directly  proportional  to  the  potential 
drop  which  is  being  measured.  A high  resistance  galvano- 
meter is  usually  one  with  a great  many  turns  of  fine  wire 
in  its  coils,  and  is  therefore  quite  sensitive  to  small  currents. 

Directions. — Connect  the  apparatus  as  shown  in  the 
diagram,  Fig.  55.  Instead  of  using  two  separate  resistance 
boxes  for  R and  S use  a single  box  with  a traveling  plug 
(a  plug  furnished  with  a binding  post)  in  place  of  the  10 
ohm  plug.  (See  Fig.  57  for  a similar  arrangement.  Here 
the  traveling  plug  is  shown  at  D.)  Connect  the  low  re- 
sistance coils  (1  to  4 ohms)  as  a shunt  to  the  galvanometer, 
the  high  resistance  coils  in  series  with  it.  Vary  the  shunt 
resistance  from  one  to  four  ohms  and  the  series  resistance 
from  4000  to  10000  ohms  as  shown  on  the  data  sheet,  and 
observe  the  deflections  in  millimeters  for  each  case  with  the 
current  direct  and  reversed.  With  the  mean  deflection  in 
each  case  calculate  the  figure  of  merit  for  a scale  distance 
of  one  meter,  remembering  that  the  scale  is  actually  only 
a half  meter’s  distance  from  the  galvanometer  coil. 

Questions. — 1. — Assuming  the  internal  resistance  of 
the  cell  to  be  a(  = 2)  ohms,  what  is  the  % error  in  your  value 
of  the  figure  of  merit  due  to  neglecting  B-\-SG/(S-{-G ) in 
your  calculation,  when  the  series  resistance  is  c(  = 4000) 
ohms? 

2.  — A galvanometer  whose  figure  of  merit  (as  defined) 
is  a(  = 10"7)  is  to  be  used  as  an  ammeter.  If  its  resistance 
is  £(  = 400)  ohms,  calculate  the  necessary  shunt  resistance 
to  make  it  read  c(  — 1.)  m.  m.  deflection  at  a meter  distance 
per  ^(  = 0.001)  ampere  in  the  main  circuit. 

3.  — A galvanometer  whose  resistance  is  <s(  = 500)  ohms, 
has  a shunt  of  b(  = 1.)  ohm,  and  a figure  of  merit  of  c(  = 10"8). 


In  General  Physics 


217 


What  series  resistance  must  be  used  with  this  apparatus  so 
that  it  may  serve  as  a voltmeter,  reading  d(  = 1.)  volt  per 
*(  = 1.)  cm.  deflection  at  a half  meter  distance. 

4. — A galvanometer  with  figure  of  merit  #(  = 2X10~7). 
and  a resistance  £(  = 2200)  ohms  is  placed  in  parallel  with  a 
circuit  having  a resistance  c(  = 5)  ohms.  If  the  galvano- 
meter reading  is  <i(  = 10)  cm.  on  a scale  at  *(  = 50)  cm. 
distance,  what  is  the  current  strength  in  the  main  circuit? 

40.  RESISTANCE  OF  A GALVANIC'CELL  BY  OHM  ’S 

METHOD 

To  determine  the  internal  resistance  of  a cell,  or  the  ratio 
of  the  e.m.f.  of  the  cell  to  the  strength  of  the  current  which 
passes  from  one  plate  to  the  other,  through  the  liquid  of 
the  cell.  (D.  442,  443,  469-476;  C.  837,  845,  846,  927,  944; 
K.  622,  623,  625;  W.  478,  480.) 

Apparatus. — Gravity  cell,  tangent  galvanometer,  re- 
sistance box,  reversing  switch,  and  connecting  wires. 

Theory  and  Method. — The  method  used  is  a simple 
application  of  Ohm’s  law  to  a circuit  containing  the  cell 
whose  resistance  is  to  be  measured  in  series  with  a tangent 
galvanometer  and  a resistance  box.  A switch  for  reversing 
the  current  in  the  galvanometer  is  also  included  in  the  circuit. 

Let  E be  the  e.m.f.  of  the  cell,  B its  internal  resistance, 
G that  of  the  galvanometer,  D its  deflection,  and  K its  re- 
duction factor.  Then  if  the  box  resistance  be  R'  the  current 
strength  will  be 

r = K tand'  = E/  (R'  + G + B ) 

If  the  box  resistance  is  now  changed  to  a value  R"  the  cor- 
responding current  will  be 

I''  = K tan  d"  = E/  ( R"  + G + B) 


218 


Manual  of  Experiments 


from  these  two  equations  we  may  eliminate  K and  E and 
obtain  an  expression  for  B in  terms  of  known  quantities. 

tand"  (R" +G)  —tand'  ( R'-\-G ) 
tand'  — tand" 

Directions. — Connect  the  apparatus  as  shown  in  the 
diagram,  Fig.  47.  Vary  the  resistance  R until  the  deflection 
is  about  30°,  and  note  the  deflection  of  both  ends  of  the  point- 
er with  current  direct  and  reversed.  Change  the  resistance 
R until  the  deflection  is  about  35°  and  note  the  deflections 
as  before.  Repeat  this  twice  more  with  angles  of  40  and  45°, 
and  55  and  60°  respectively,  and  calculate  B using  the  mean 
deflections. 

Questions. — 1. — Explain  in  detail  why  this  experiment 
is  adaptable  to  cells  which  do  not  polarize  rapidly,  and  to 
no  others? 

2.  — Supposing  a tangent  galvanometer  were  not  avail- 
able, derive  the  expression  for  the  internal  resistance  of  a 
cell  when  using  a shunted  D’Arsonval  galvanometer  as  the 
ammeter.  (See  Exp.  39.) 

3.  — A battery  is  in  series  with  a resistance  and  a 
D’Arsonval  galvanometer  of  resistance  ^(  = 200)  ohms 
which  is  shunted  by  a coil  of  b{  = 2)  ohms  resistance.  When 
R is  c(  = 5000)  ohms  the  deflection  is  d(  = 10  cm.),  and  when 
R is  ^(  = 9000)  ohms,  the  deflection  is  f(  = 5.6)  cm.  Find 
the  average  resistance  of  the  battery. 

41.  POTENTIAL  DIFFERENCE  AT  THE  TERMI- 
NALS OF  A CELL  AS  A FUNCTION  OF  THE 
EXTERNAL  RESISTANCE 

To  show  that  the  potential  difference  at  the  terminals 
of  a galvanic  cell  depends  upon  the  external  resistance,  and 
that  for  a high  value  of  this  resistance  the  potential  differ- 


In  General  Physics 


219 


ence  approaches  a fixed  value,  the  e.m.f.  of  the  cell.  Also 
to  determine  the  e.m.f.  and  the  internal  resistance  of  the  cell, 
by  a graphical  method.  (D  442,  443,  452;  G.  844,  845,  854; 
K.  641,  642;  W.  486,  490.) 

Apparatus. — D’Arsonval  galvanometer  and  shunt,  grav- 
ity cell,  resistance  box  with  traveling  plug,  reversing  switch, 
eight  connecting  wires. 

Theory  and  Method. — The  e.m.f.  of  a cell  was  defined 
as  the  difference  of  potential  between  its  terminals  on  open 
circuit , that  is  when  no  current  is  flowing  through  it.  If 
now  the  circuit  external  to  the  cell  be  closed  through  a re- 
sistance which  is  not  infinite,  a current  will  flow  through  the 
circuit,  and  the  potential  difference  at  the  terminals  of  the 
cell  will  no  longer  equal  its  e.m.f.  The  current  strength  in 
the  circuit  will  equal  the  total  electromotive  force  in  the 
circuit  divided  by  the  sum  of  all  the  resistances,  and  in 
this  case  the  circuit  is  completed  through  the  internal 
resistance  of  the  cell.  Hence  if  E be  the  e.m.f.  of  the  cell, 
B its  internal  resistance,  R the  external  resistance,  and  I 
the  current,  we  have  by  Ohm’s  law 

I = E/(R+B) 
or 

E = IR+IB  (1) 

The  difference  of  potential  P.D. , between  the  terminals 
which  equals  the  drop  of  potential  IR  through  i?,  is  less  than 
E by  the  drop  of  potential  e(  = IB ) through  the  internal  re- 
sistance B,  or 

P.D.  = E — e 

It  is  evident  that  as  R is  increased,  I (and  therefore  e also) 
will  become  smaller  and  smaller  so  that  the  value  of  P.D. 


220 


Manual  of  Experiments 


will  gradually  approach  that  of  E.  When  R is  infinitely 
great , or  practically  when  there  is  an  open  circuit, 

P.D.=E 

For  equal  values  of  external  and  internal  resistance  the  drop 
of  potential  through  the  two  circuits  is  the  same,  and  equa- 
tion (1)  becomes  E = 2lR,  therefore 

P.D.  =e  = E/2 

Bearing  these  facts  in  mind  and  plotting  a curve  with 
values  of  R as  abscissae  and  corresponding  values  of  P.D. 
as  ordinates,  we  may  readily  determine  E as  the  limiting 
value  of  P.D.  as  R becomes  infinite,  (i.  e.  as  the  ordinate  of 
the  asymptote  to  the  curve.)  From  the  same  curve,  B 
may  be  determined  by  measuring  the  abscissa  of  that  point 
on  the  cur  ye  for  which  P.D.=E/2  (i.  e.  that  point  which 
lies  half  way  between  the  asymptote  and  the  axis  of  abscissae.) 

If  we  plot  reciprocals  of  P.D.  and  of  R as  coordinates 
the  curve  will  be  a straight  line.  The  relation  between 
P.D.  and  R may  be  found  from  equation  (1).  The  ratio  of 
E to  P.D.  is  evidently 

E/P.D.  = I(R+B)/IR 
or, 

E B 

=1+— 

P.D  R (2) 

If  we  put  1/R  equal  to  zero  in  equation  (2),  then 
E/P.D.  = 1,  or 

E = P.D. 

But  the  point  on  the  curve  whose  abscissae  is  1/R  = 0,  has  for 
its  ordinate  1/P.D.  the  portion  of  the  axis  of  ordinates  in- 
tercepted by  the  curve.  Therefore  the  e.m.f.  of  the  cell  will 
equal  the  value  of  P.D.  whose  reciprocal  is  represented  by  the 
intercept  of  the  curve  on  the  axis  of  ordinates. 


In  General  Physics 


221 


If  we  put  1/P.D.  equal  to  zero  in  equation  (2),  then 
l-\-B/R  = 0,  or 

B=-R 

But  the  abscissa  1/R corresponding  to  1/P.D.  =0  is  the  inter- 

\ 

cept  of  the  curve  on  the  axis  of  abscissae,  and  since  it  lies 
to  the  left  of  the  origin  it  will  have  a negative  sign,  so  that 
— R will  be  positive.  Therefore  the  internal  resistance  of 
the  cell  will  equal  the  value  of  R whose  reciprocal  is  represented 
by  the  intercept  of  the  curve  on  the  axis  of  abscissae. 

In  order  to  measure  the  drop  of  potential  P.D.  through 
R , a shunted  galvanometer  with  a high  resistance  Ri  in  series 
as  shown  in  figure  56,  is  used  as  a voltmeter.  (See  Exp.  39). 
The  potential  difference  between  the  terminals  C and  D or 
between  the  terminals  of  the  cell  B will  equal  the  drop  of 
potential  through  the  galvanometer  circuit,  or  its  resistance 
times  the  strength  of  the  main  current  in  the  galvanometer 


circuit.  If  I g is  the  galvanometer  current,  the  current  in 
Ri  will  be  /g  (S+G)/S.  The  total  resistance  will  be 
R i + (SG/(S  + G)),  but  if  Ri  is  large,  say  10,000  ohms, 
SG/  (S  + G)  will  be  negligible  in  comparison.  Then  the 
drop  of  potential  through  CD  will  be 

r=RJe  «S+G)/S) 

If  / g be  taken  as  the  galvanometer  current  in  amperes 
necessary  to  produce  a deflection  of  one  mm.  (the  figure 
of  merit),  V will  be  the  potential  difference  in  volts  between 


222 


Manual  of  Experiments 


C and  D for  one  mm.  deflection.  For  a deflection  of  d 
mm.,  the  number  of  volts  will  be 

(S+G) 

P.D.  =dXV.  = dRih 

S 

Directions. — The  apparatus  is  arranged  as  shown  in 
figure  56.  The  galvanometer  has  a low  resistance  shunt  of 
about  4 ohms  and  a resistance  of  10,000  ohms  in  series. 
The  series  resistance  Rx  and  the  external  resistance  R are 
included  in  the  same  box,  connection  being  made  to  a travel- 
ing plug  at  D.  This  is  shown  in  figure  57,  where  the  box 
terminals  are  marked  F and  C to  correspond  with  figures 
56,  and  the  traveling  plug  is  marked  D.  The  figure  shows 
the  plugs  removed  to  place  10000  ohms  in  series  with  the  gal- 
vanometer. Vary  the  resistance  CD  from  infinity  to  one 
ohm  as  shown  on  the  data  sheet  and  note  the  corresponding 


Fig.  57 


In  General  Physics 


223 


deflections  of  the  galvanometer.  To  make  R infinitely 
large,  disconnect  the  two  wires  leading  from  the  terminal  C 
of  the  resistance  box  to  the  galvanometer  and  switch,  but 
keep  these  wires  joined  together.  From  the  value  of  the 
figure  of  merit  of  the  galvanometer,  the  values  of  P.D.  for 
different  values  of  R are  to  be  calculated.  The  two  curves 
mentioned  above  are  to  be  plotted  and  the  value  of  E and 
of  B should  be  obtained  from  each  of  them. 

Questions. — 1. — Plot  the  two  curves  described  in  the 
above  experiment.  What  are  the  e.m.f.  and  internal  re- 
sistance of  the  cell  as  determined  from  each  curve? 

2.  — If  you  used  a galvanometer,  whose,  resistance  is 
#(  = 250.)  ohms,  with  a shunt  of  b(  = 4.)  ohms  and  a series 
resistance  of  c(  = 4000.)  ohms,  what  is  the  % error  in  your 
calculated  value  of  P.D.  due  to  neglecting  SG/(S+G)? 

3.  — A shunt  wound  dynamo  has  its  field  coils  connected 
between  the  terminals,  in  parallel  with  the  external  circuit. 
If  the  armature  (internal)  resistance  of  such  a dynamo  is 

= 0.0404)  ohm  and  its  field  resistance  is  b(  = 3.75)  ohms, 
• what  must  be  the  P.D.  at  the  terminals  to  properly  light 
c(  = 200)  incandescent  lamps  in  parallel,  each  lamp  requiring 
^(  = 0.5)  ampere  at  ^(  = 100)  volts,  the  resistance  of  the 
connecting  leads  being  /( = 0.1)  ohm?  What  e.m.f . must 
be  generated  in  the  armature  under  these  conditions? 

4.  — A certain  battery  with  #(  = 4.5)  volts  e.m.f.  will 
send  b{  = 1.5)  amperes  through  an  external  resistance  of 
c(  = l.)  ohm.  Through  what  resistance  will  it  send  ^(  = 0.01) 
ampere? 


224 


Manual  of  Experiments 


42.  ELECTROLYSIS  AND  THE  COPPER 

VOLTAMETER 

To  measure  a current  strength  with  the  copper  volta- 
meter, and  by  passing  this  current  through  a tangent  gal- 
vanometer to  determine  the  horizontal  intensity  of  the 
earth ’s  field  from  its  reduction  factor.  (D.  462-468;  G. 
939-941;  953-956;  K.  608-620;  W.  539,  540,  478.) 

Apparatus. — Voltameter  jar  with  copper  sulphate  solu- 
tion, two  copper  anodes,  one  copper  kathode,  electrode 
holder,  tangent  galvanometer,  reversing  switch,  fine  balance 
and  weights,  fine  sand  paper,  filter  paper,  connecting  wires, 
and  110  volt  circuit  with  lamp  resistance. 


Theory  and  Definitions. — A solution  through  which 
a current  of  electricity  is  conducted  by  a process  similar  to 
convection,  is  termed  an  electrolyte,  and  the  process  is 
called  electrolysis.  The  dissolved  substance  is  dissociated 
by  the  act  of  solution;  that  is,  some  of  its  molecules  separate 
into  two  parts  called  ions  and  the  weaker  the  solution  the 
greater  is  the  number  of  free  ions.  These  ions  are  the  con- 
stituent atoms  or  groups  of  atoms , one  bearing  a positive 
charge  of  electricity , the  other  a negative  charge.  A current 
of  electricity  may  be  passed  through  the  solution  between 
two  conductors  dipping  into  it.  These  terminal  conductors 
are  called  electrodes , and  that  one  by  which  the  current  enters 
the  solution  is  known  as  the  anode , the  one  by  which  the  current 


In  General  Physics 


225 


leaves  it , the  kathode.  The  electrolyte  and  the  electrodes 
form  an  electrolytic  cel]. 

When  a current  of  electricity  passes  through  an  elec- 
trolytic cell  the  positively  charged  ions  move  in  the  direction 
of  the  current , toward  the  kathode.  They  are  therefore  called 
kations.  The  negatively  charged  ions  move  toward  the  anode 
and  are  called  anions.  These  anions  and  kations  alternately 
separate  and  combine  with  each  other,  and  in  this  process 
some  of  them  will  appear  at  the  electrodes,  where  they  gen- 
erally give  up  their  charge.  They  may  adhere  to  the  elec- 
trode as  in  the  case  of  most  metals,  or  may  escape  from  the 
solution,  as  in  the  case  of  gases  of  less  density  than  the  latter, 
or  they  may  enter  into  secondary  relations  with  the  sur- 
rounding solution. 

The  laws  of  electrolysis,  known  as  Faraday ’s  laws,  may 
be  stated  as  follows: — The  mass  of  the  ions  liberated  at  either 
electrode  is  proportional  1st.  to  the  quantity  of  electricity  which 
passes  through  the  solution.  2nd.  to  the  chemical  equivalent 
of  the  ion. 

The  quantity  of  electricity  in  coulombs  equals  the  cur- 
rent strength  / in  amperes  multiplied  by  the  time  t in  seconds 
during  which  the  current  flows.  The  chemical  equivalent 
of  the  ion  is  its  atomic  weight  w divided  by  its  valency  v in 
the  electrolyte  used.  Hence  the  mass  m of  ions  liberated  is 

m = k—lt  = zlt 

V 

The  quantity  z = kw/v  is  a constant  for  any  one  electrolyte 
and  is  called  the  electrochemical  equivalent  of  the  ion.  If 
we  put  It  equal  to  unity  we  see  that  m equals  kw/v,  or  the 
electrochemical  equivalent  may  be  defined  as  the  mass  of  the 
ion  set  free  by  unit  quantity  of  electricity. 

Taking  the  atomic  weight  and  valency  of  hydrogen  both 


226  Manual  of  Experiments 

equal  unity,  its  electro  chemical  equivalent  is  numerically 
equal  to  k.  That  is  to  say,  the  proportionality  factor  k is 
numerically  equal  to  the  mass  of  hydrogen  ions  in  grams 
liberated  by  one  coulomb  of  electricity.  But  when  the 
atomic  weight  of  oxygen  is  used  as  the  base  of  the  system  of 
weights,  this  relation  no  longer  holds  true  for  the  chemical 
equivalent  of  hydrogen  instead  of  being  unity  will  be  1.008. 

The  value  of  k may  be  determined  by  electrolysis  of  a 
solution  in  which  hydrogen  is  liberated,  but  is  more  accu- 
rately obtained  from  a determination  of  the  electrochemical 
equivalent  of  silver  from  a silver  nitrate  solution.  The 
value  of  k is  found  to  be  1.044X(10-5)  grams  per  coulomb. 

Knowing  the  electrochemical  equivalent  of  an  ion,  we 
may  measure  the  mass  of  it  liberated  in  a given  time  and  so 
determine  the  current  strength  which  liberates  it.  An 
electrolytic  cell  used  for  this  purpose  is  called  a coulomb- 
meter  or  voltameter.  The  copper  voltameter  may  consist 
of  two  copper  plates  in  a solution  of  copper  sulphate.  (Den- 
sity 1.15  to  1.18.  20  to  25  grams  of  copper  sulphate  crystals 

in  80  grams  of  water,  with  1%  of  concentrated  sulphuric 
acid  added).  The  solution  dissociates  into  copper  kations 
and  sulphion  (SCh)  anions.  The  copper  anode  goes  into 
solution  by  combining  with  the  anions  liberated  in  its  neigh- 
borhood, so  that  the  solution  surrounding  it  becomes  more 
concentrated.  The  ions  liberated  at  the  kathode  are  de- 
posited upon  the  latter,  the  solution  in  its  neighborhood  be- 
coming less  concentrated.  Taking  the  atomic  weight  of 
copper  as  63.64  and  its  v?lency  as  2,  its  electrochemical 
equivalent  will  be  0.0003294  grams  per  ampere  second. 
The  gain  in  mass  of  the  kathode  in  a given  time  may  be  ac- 
curately determined,  and  from  it  we  can  readily  calculate 
the  current  strength. 


In  General  Physics 


227 


Directions. — Sandpaper  the  lugs  on  the  electrodes  and 
the  clips  on  the  holder  so  that  they  make  good  contact  with 
each  other,  and  sandpaper  the  electrodes  so  that  they  pre- 
sent a fresh,  unoxidized  surface  toward  each  other.  Clamp 
the  lugs  in  the  holder,  the  thin  kathode  being  symmetrically 
placed  between  the  two  heavier  anodes,  so  that  the  plates 
are  parallel  to  and  level  with  each  other.  Place  them  in  the 
jar  and  fill  it  with  copper  sulphate  solution  up  to  the  lugs 
on  the  plates.  Connect  the  apparatus  as  shown  in  figure 
58,  so  that  the  current  from  the  source  B passes  through 
the  electrolytic  cell  C in  series  with  the  tangent  galvano- 
meter G , and  can  be  reversed,  in  the  galvanometer  only,  by 
the  reversing  switch.  Connect  to  the  smallest  coil  of  the 
galvanometer  (middle  terminals)  so  that  a large  current  may 
be  used.  The  current  strength  is  governed  by  the  lamp  re- 
sistance. The  positive  terminal  of  the  source  of  current 
must  be  connected  to  tfae  anodes.  The  terminals  of  B should 
be  marked  + and  — , but  if  necessary,  the  direction  of  the 
current  may  be  determined  by  noting  on  which  plate  the 
current  deposits  copper  in  a preliminary  trial,  or  by  noting 
the  deflection  of  a compass  needle  placed  under  the  connect- 
ing wires.  * 

Allow  the  current  to  flow  for  five  minutes  in  order  to  give 
the  kathode  a preliminary  coating.  Then  remove  the 
kathode,  wash  it  carefully  by  flowing  soft  water  over  it,  and 
cJryT>y  pressing  gently  between  sheets  of  filter  paper  without 
rubbing.  The  copper  will  not  adhere  to  the  plates  unless 
they  are  perfectly  free  from  oil  or  grease.  Those  parts  of 
the  plates  which  are  to  be  immersed,  mud  therefore  not  be 
touched  by  the  hand  nor  laid  on  a dusty  table  or  balance 
pan.  When  quite  dry  the  kathode  should  be  carefully 
weighed  on  the  fine  balance  and  the  mass  recorded.  Replace 
the  plates  in  the  cell  and  close  the  circuit,  noting  the  exact 


228 


Manual  of  Experiments 


time  when  the  current  starts  to  flow.  Read  the  deflections 
of  both  ends  of  the  galvanometer  pointer  and  at  the  end  of 
five  minutes  reverse  the  current.  While  throwing  over  the 
reversing  switch  the  current  will  be  interrupted.  It  is  there- 
fore necessary  to  reverse  as  rapidly  as  possible.  Repeat  the 
procedure,  reversing  the  current  and  reading  the  galvano- 
meter deflections  at  the  end  of  every  five  minutes  until  the 
current  has  been  flowing  for  one  hour.  Then  open  the  switch^ 
remove  the  kathode,  and  wash,  dry  and  weigh  it  as  before, 
and  record  the  mass.  The  copper  sulphate  solution  should 
be  returned  to  the  supply  jar. 

While  the  copper  is  being  deposited,  the  number  of  turns 
of  wire  on  the  galvanometer  coil,  and  the  mean  radius  of  a 
turn  should  be  recorded,  and  from  these  the  galvanometer 
constant  G calculated.  Determine  the  mean  current  strength 
from  the  change  in  mass  of  the  kathode.  As  the  copper 
plates  dissolve  in  the  electrolyte  at  a definite  rate  per  second 
per  unit  area  of  electrode  surface,  it  is  usually  necessary  to 
allow  for  this  in  calculating  the  amount  of  copper  deposited 
but  for  the  small  plates  used  and  the  short  time  of  deposit, 
this  correction  is  negligible.  Note  the  tangent  of  the  mean 
angle  of  deflection  and  calculate  K , the.reduction  factor  of 
the  galvanometer.  The  horizontal  component  of  the  earth ’s 
field  is  calculated  from  the  galvanometer  constant  and  the 
reduction  factor. 

Questions. — 1. — What  is  the  advantage  of  the  tangent 
galvanometer  over  the  D’Arsonval  for  this  experiment? 
( b ) — Why  is  the  switch  reversed  every  five  minutes? 

2. — A current  of  a(  = 1)  ampere  will  decompose  how  much 
b(=  water)  in  *(  = 1)  hour  if  the  atomic  weight  of  d{  = 
hydrogen)  is  *(  = 1.008)? 

— A voltameter  containing  a solution  of  a salt  of  a(  = 
nickel)  with  a resistance  of  &(  — M)  ohm  between  its  ter- 


In  General  Physics 


229 


minals,  has  a constant  potential  difference  of  c(  = 2)  volts 
maintained  between  them.  How  much  of  a will  be  deposited 
in  d{  = 15)  minutes? 

4. — A copper  voltameter  is  in  series  with  a tangent 
galvanometer.  A current  passing  through  it  deposits  a 
( = 0.912)  grams  of  copper  on  the  kathode  in  b(  = 20)  minutes, 
and  the  galvanometer  deflection  is  c(  = 30)°.  What  is  the 
reduction  factor  of  the  galvanometer? 


43 . ELECTROMAGNETIC  INDUCTION 


To  study  electromagnetic  induction  qualitatively  (D. 
499-505;  G.  957-963;  K.  710-726;  W.  516-519.) 

Apparatus. — A primary  and  secondary  coil  of  wire 
ballistic  D’Arsonval  galvanometer,  reversing  switch,  horse 
shoe  magnet,  iron  core,  110  volt  circuit  with  lamp  resistance 

Theory  and  Definitions. — Whenever  the  number  o 
lines  of  force  threading  through  a closed  circuit  is  altered,  a 
current  of  electricity  is  induced  in  the  circuit.  This  phenome- 
non is  called  electromagnetic  induction.  Whenever  the  in- 
duction results  from  relative  motion  between  the  circuit  and  a 
source  of  magnetic  lines  of  force,  Lenz’s  law  states  that  the 
induced  current  is  always  in  such  a direction  that  it  opposes 
by  its  electromagnetic  action , the  motion  which  produced  it. 

The  change  in  the  number  of  lines  of  force  may  be  due  to 

(a)  Relative  motion  between  the  closed  circuit  and  a 
magnet; 

( b ) Relative  motion  between  the  closed  circuit  and  a 
second  closed  circuit  through  which  a steady  current  is 
flowing; 


230 


Manual  of  Experiments 


(c)  A change  in  the  strength  of  *the  current  flowing 
through  the  second  closed  circuit  when  both  circuits  are  at 
rest. 

In  cases  (b)  and  (c)  that  circuit  in  which  the  current  is 
induced  is  called  the  secondary  circuit,  the  other  is  the  pri- 
mary circuit.  The  current  lasts  only  during  the  time  that 
the  change  in  the  number  of  lines  is  taking  place,  and  a 
ballistic  galvanometer  must  be  used  to  indicate  the  total 
flow  of  current. 

It  is  sometimes  more  convenient  to  think  of  the  current 
as  induced  in  a conductor  by  cutting  across  lines  of  force. 
In  this  case  the  direction  of  the  induced  current  is  given  by 
Fleming’s  rule.  Place  the  thumb,  forefinger  and  middle 
finger  of  the  right  hand  mutually  perpendicular  to  each  other. 
Then  if  the  thumb  be  pointed  in  the  direction  of  motion  and  the 
forefinger  in  the  direction  of  the  lines  of  force,  the  middle  finger 
will  point  in  the  direction  of  the  induced  current. 

The  permeability  of  iron  is  greater  than  that  of  air;  that 
is,  for  the  same  field  strength,  there  will  be  more  lines  of 
force  in  a given  cross  section  of  iron  than  in  the  same  cross 
section  of  air  at  the  same  point  in  the  field.  Hence  if  we  place 
an  iron  core  through  one  or  both  of  the  circuits,  the  change 
in  the  number  of  lines  of  force  in  the  secondary  will  be 
greater  than  before. 


Directions. — Connect  the  apparatus  as  shown  in  the 
diagram,  Fig.  59.  The  secondary  coil  B is  fixed,  and  con- 
nected to  the  galvanometer  G.  The  primary  coil  A is  mov- 


In  General*  Physics 


231 


able  and  may  be  turned  about  a vertical  axis  or  may  be  moved 
to  different  distances  from  B.  The  coil  A is  connected  to 
the  source  of  current  B through  the  reversing  switch  K. 

(1)  Relative  motion  between  a magnet  and  a closed  cir- 
cuit.— Bring  the  horse-shoe  magnet  rapidly  astraddle  of  the 
coil  B and  note  the  direction  of  the  deflection  of  the  gal- 
vanometer. When  the  galvanometer  coil  has  returned  to 
its  position  of  rest,  rapidly  withdraw  the  magnet  from  the 
coil  and  note  the  deflection.  Repeat  this  with  the  magnet 
poles  reversed. 

Relative  motion  between  two  circuits. 

(2)  Primary  and  secondary  circuits  approaching  or  re- 
ceding from  each  other. — Place  the  primary  coil  A parallel  to 
the  secondary  coil  at  the  opposite  ends  of  the  supporting 
board,  and  close  the  switch  so  that  the  current  flows  through 
A.  Move  A suddenly  up  to  B and  note  the  direction  of 
deflection.  When  the  galvanometer  is  again  at  rest,  with- 
draw A and  note  the  deflection.  Reverse  the  current  and 
repeat. 

(3)  Rotation  of  primary  circuit. — Insert  the  brass  pin 
through  the  holes  in  the  base  of  the  coil  and  the  supporting 
board  and  place  A parallel  to  B.  Rotate  A suddenly 
through  90°  and  note  the  amount  and  direction  of  deflection. 
When  the  galvanometer  is  again  at  rest,  rotate  A through  a 
further  90°  in  the  same  direction  and  at  approximately  the 
same  speed,  and  note  the  direction  and  amount  of  deflec- 
tion. Compare  the  two  deflections.  Then  bring  A back  to 
its  original  position  and  when  the  galvanometer  is  at  rest, 
rotate  it  in  the  same  direction  as  before  and  with  approxi- 
mately the  same  speed,  but  through  180°.  Allow  the  gal- 
vanometer to  come  to  rest  and  rotate  A through  another 
180°  as  before.  Compare  the  two  deflections  in  this  case 
in  direction  and  amount. 


232  Manual  of  Experiments 

(4)  Variable  current  in  the  primary . — Bring  A close  up 
to  and  parallel  to  B , and  when  the  galvanometer  is  at  rest, 
suddenly  close  the  primary  circuit  and  note  the  deflection. 
After  the  galvanometer  is  again  at  rest,  suddenly  open  the 
switch.  Repeat  with  the  switch  reversed.  Compare  the 
four  deflections. 

(5)  Primary  at  different  angles. — Place  A on  the  brass 
pin  and  note  the  deflections  on  breaking  the  circuit  when 
the  plane  of  coil  A makes  the  following  angles  with  the  plane 
of  B : — 0°,  45°,  60°,  75°,  90°,  105°,  120°,  135°,  180°.  Plot 
a curve  having  angles  as  abscissae  and  deflections  as  ordinates. 

(6)  Effect  of  iron  core. — With  A and  B parallel  and 
about  3 cm.  apart,  note  the  deflection  when  the  current  is 
broken.  Put  an  iron  core  through  the  center  of  both  coils 
and  repeat.  Compare  the  deflections. 

Questions. — 1. — With  the  data  obtained  in  section  five 
of  the  above  experiment  plot  a curve  having  angles  between 
the  planes  of  primary  and  secondary  coils  as  abscissae  and 
corresponding  deflections  of  the  galvanometer  as  ordinates. 

2. — Suppose  a circular  metal  ring  to  rotate  clockwise 
about  an  axis  through  its  center,  perpendicular  to  the  paper 
but  parallel  to  its  own  plane,  in  front  of  the  north  pole  of  a 
permanent  magnet  lying  in  the  plane  of  the  paper.  Draw 
four  diagrams  showing  the  ring  in  its  four  quadrants,  and 
indicate  on  each  the  direction  of  the  induced  current,  and 
that  of  the  lines  of  force  which  it  sets  up  in  the  ring. 

Note. — Draw  a section  through  the  center  of  the  coil 
parallel  to  the  plane  of  the  paper,  representing  the  cut 
ends  of  the  coil  by  circles.  A dot  or  dash  in  one  of  these 
circles  represents  an  approaching  current  (point  of  an 
arrow);  a cross  represents  a receding  current  (tail  of  the 
arrow.) 


In  General  Physics 


233 


3.  — Explain  how  Lenz's  law  applies  in  each  case  (by  the 
aid  of  the  north  and  south  faces  of  the  ring,)  or  how  Fleming 's 
rule  applies. 

4.  — At  a place  where  the  vertical  component  of  the  earth 's 

magnetic  field  is  a(  = 0.50),  a horizontal  coil  of  wire  of 
&(  = 10)  turns,  with  diameter  c(  = 3)  meters,  and  resistance 
d{  = 2 )ohms,  is  suddenly  pulled  out  into  a straight  line 
loop  in  e(  = y2)  second.  Find  a)  the  induced  e.m.f.;  b ) the 
average  current  in  amperes;  c) th^  total  flow  of  electricity 
in  coulombs;  and  d)  the  energy  used  up.  g 4/ 

44.  JOULE'S  LAW  AND  THE 
ELECTRO-CALORIMETER 

To  determine  Joule's  equivalent  by  the  electro-calori- 
meter. (D.  458-461;  G.  403,  858-861;  K 654-657;  W.  493, 
494.) 

Apparatus. — Calorimeter  with  high  resistance  coil, 
tangent  galvanometer,  reversing  switch,  source  of  current, 
thermometer,  coarse  balance  and  weights,  P.  0.  box  bridge, 
dry  cell,  connecting  wires  and  watch  or  clock. 

Definitions  and  Theory. — Whenever  a current  of 
electricity  flows  through  a conductor  it  generates  heat.  If 
the  difference  of  potential  between  the  terminals  of  the  con- 
ductor be  V the  energy  expended  when  unit  quantity  of 
electricity  flows  through  it  will  be  V units  by  definition  of 
difference  of  potential.  If  in  the  time  T,  q units  flow 
through  the  conductor,  the  work  done  will  be 

W = qV 

But  if  i be  the  current  strength,  q = iT  and  if  r be  the  re- 


234 


Manual  of  Experiments 


sistance  of  the  conductor,  V = ir,  by  Ohm  ’s  law,  when  ce 

W=  i2rT 

When  i,  r and  t are  measured  in  C.  G.  S.  absolute  units,  the 
work  done  will  be  expressed  in  ergs,  but  if  C be  the  current 
in  amperes  and  R the  resistance  in  ohms,  since  one  ampere 
equals  lO-1  C.  G.  S.  units  of  current  and  one  ohm  equals 
IQ9  C.  G.  S.  units  of  resistance,  the  work  done  will  be 

W=  107  G2RTergs  = C2RT  joules 

the  joule  being  defined  as  a unit  of  energy  equal  to  107  ergs. 
Then  if  one  ampere  flows  through  a resistance  of  one  ohm 
for  one  second  the  work  done  will  be  one  joule.  If  we  wish 
to  express  the  energy  expended  as  heat  in  calories  we  must 
divide  by  the  mechanical  equivalent  of  heat,  /,  in  ergs  per 
calorie,  whence 

107  C2  R T 

II  = calories. 

*/ 

This  is  known  as  Joule’s  law. 

By  passing  a current  of  electricity  through  a coil  of  wire 
immersed  in  a water  calorimeter  and  measuring  the  strength 
of  current,  resistance  of  coil,  time  of  flow  and  heat  generated, 
we  may  determine  Joule’s  equivalent,  /,  by  the  above  for- 
mula. If  however,  we  assume  the  value  of  /,  we  may  de- 
termine the  resistance  of  the  wire  or  the  strength  of  the 
current. 

Directions. — Connect  the  coil  of  calorimeter  and 
tangent  galvanometer  in  series  with  the  source  of  current 
and  arrange  the  switch  so  that  the  current  may  be  reversed. 
Before  starting,  measure  the  resistance  of  the  coil  and  weigh 
the  calorimeter  and  stirrer.  Cool  some  water  to  about  10 
degrees  below  the  room  temperature  and  fill  the  calorimeter 


In  General  Physics 


235 


about  three-fourths  full.  Determine  the  weight  of  water 
added.  Note  the  temperature  of  water,  and  at  the  instant 
of  closing  the  circuit  note  the  exact  time  and  observe  the  gal- 
vanometer deflection.  At  the  end  of  every  five  minutes 
reverse  the  current  and  observe  the  deflection.  Keep  the 
water  well  stirred.  Watch  the  thermometer  and  when  the 
temperature  has  risen  to  about  10  degrees  above  that  of 
the  room,  open  the  circuit,  carefully  observing  the  time  and 
temperature  at  the  instant  of  opening.  Redetermine  the 
resistance  of  the  coil,  and  find  the  mean  of  these  two  deter- 
minations. Calculate  the  water  equivalent  of  the  calori- 
meter cup  and  stirrer.  Determine  the  heat  generated  from 
the  change  in  temperature.  Average  the  galvanometer  de- 
flections and  determine  the  current  strength.  With  this 
data  determine  the  value  of  /. 

Questions. — 1. — Given  two  conductors  of  the  same 
material  and  length  but  of  different  cross  sections.  In 
which  will  the  energy  loss  per  unit  time  by  heating  be  the 
greater  a)  when  they  are  in  series?  b)  When  they  are  in 
parallel?  Give  reasons  for  your  answers.  (Notice  that 
C2R  = E2/R.) 

2.  — If  coal  costing  <s(  = 4)  dollars  per  ton  develops 
£(  = 8000)  calories  of  heat  per  gram,  and  electricity  costs 
c(  = 10)  cents  per  kilowatt  hour,  how  does  the  cost  of  heating 
by  coal  compare  with  that  of  heating  by  electricity? 

3.  — An  electric  motor  of  a(  = 2)  kilowatt  capacity, 
operating  at  £(  = 100)  volts,  receives  current  through  a pair 
of  copper  wires  from  a station  which  is  c(==l)  kilometer 
away.  If  the  energy  loss  per  second  due  to  heating  in 
transmission  may  be  only  ^(  = 1)  % of  the  full  capacity 
of  the  motor,  what  must  be  the  diameter  of  the  wire? 


236 


Manual  of  Experiments 


4. — 4f  the  electricity  in  problem  3 were  transmitted  at 
^(  = 2000)  volts/  and  the  same  loss  of  energy  by  heat  were 
allowed,  how  much  would  be  saved  on  the  cost  of  the  trans- 
mission lines  when  copper  is  at  /(  = 15)  cents  per  pound? 


u 


In  General  Physics 


237 


Experiments  in  Light. 

PHOTOMETRY. 

Definitions  and  Theory. — (D.  627-629,  712,  713;  G. 
519-521;  K .793-800;  W.  358-361.) 

The  comparison  of  intensities  of  illumination,  is  called 
photometry  and  a photometer  is  a device  for  making  such 
comparisons.  It  may  be  shown  that  the  intensity  of  illu- 
mination, or  amount  of  light  per  unit  area,  varies  inversely 
as  the  square  of  the  distance  from  the  source.  That  is  if 
I'  be  the  intensity  at  a unit  distance  from  the  source,  the 
intensity  at  a distance  r will  be  1 = 1'/ (V2).  Photometers  are 
usually  so  arranged  that  the  intensity  of  illumination  pro- 
duced on  a screen  by  the  source  to  be  tested,  may  be  made 
equal  to  the  intensity  (on  the  same  screen)  from  the  standard 
with  which  it  is  to  be  compared.  This  may  be  done  by  keep- 
ing the  two  sources  fixed  in  position  on  the  photometric 


Fig.  60 


bench  and  moving  the  screen  on  the  bench  between  them, 
until  the  intensity  of  illumination  is  the  same  from  both 
sources.  See  Fig.  60.  One  of  the  first  standards  to  be  used 
was  a candle  made  according  to  special  specifications  and 
burning  at  a definite  rate.  The  intensity  of  illumination  is 
therefore  often  expressed  in  1 ‘ candle-feet,  ’ ’ that  is,  in  terms 


238 


Manual  of  Experiments 


of  the  number  of  standard  candles  which  would,  at  a distance 
of  one  foot  from  the  center  of  the  flame  (in  the  horizontal 
plane  through  this  point),  give  the  same  intensity  as  the 
source  in  question  at  the  same  distance.  Suppose  we  are 
comparing  two  sources  of  illumination,  P and  Q , and  that 
when  the  screen  is  equally  illuminated  by  both,  its  distances 
from  P and  Q are  p and  q respectively.  Let  P be  the  in- 
tensity of  illumination  one  foot  from  P and  /"  the  intensity 
one  foot  from  Q.  Then  the  intensity  at  the  screen  will  be 

/'  I"  P 2 

/ = — = —,  or  /'  = ■/" 

p2  q2  q2 

I'  and  /"  may  be  termed  the  illuminating  power  of  the 
two  sources  and  if  Q is  a standard  candle,  /"  will*  be  one 
candle  power  and  P will  be  expressed  in  terms  of  candle 
power.  Various  devices  are  used  to  determine  just  when  the 
intensities  from  both  sources  are  the  same.  In  the  Bunsen 
photometer  a screen  of  white  paper  with  a grease  spot 
( h , Fig.  60)  at  the  center  has  its  two  faces  each  illuminated 
by  one  of  the  two  sources.  This  spot  is  translucent  and 
therefore  appears  darker  than  the  surrounding  paper  by 
reflected  light  and  lighter  by  transmitted  light.  When  the 
two  faces  are  equally  illuminated  the  grease  spot  will  almost 
disappear,  for  then  the  loss  in  amount  of  light  reflected  from 
the  spot  on  one  side  will  be  compensated  by  the  light  trans- 
mitted from  the  other  side  of  the  screen.  In  order  to  make 
both  sides  of  the  screen  visible  at  the  same  time,  two  plane 
mirrors,  c and  d,  are  set  at  an  angle  with  it.  It  is  advisable 
to  view  both  images  in  the  mirrors  with  the  same  eye,  for 
the  two  eyes  are  seldom  able  to  judge  equality  of  intensity 
with  the  same  degree  of  accuracy. 

If  a transparent  or  translucent  screen  be  interposed  be- 
tween the  source  and  the  photometer  screen,  the  amount 


In  General  Physics 


239 


of  light  which  passes  through  will  depend  upon  the  thickness 
of  the  screen.  Let  a be  the  fractional  part  of  the  light 
which  passes  through  unit  thickness  of  the  screen.  Then 
if  1 be  the  intensity  before  passing  through  the  screen,  the 
intensity  after  passing  through  a screen  of  thickness  of  n 
units  will  be  IoP.  For  consider  the  screen  to  be  made  up 
of  layers  of  unit  thickness.  The  intensity  after  passing 
through  the  first  layer  will  be  / a,  and  only  the  fraction  a 
of  this  amount,  or  /n2,  will  pass  through  the  second  layer: 
only  the  fraction  a of  this  amount,  or  /a3,  will  pass  through 
the  third  layer,  etc.  Let  the  intensity  of  the  source  P 
through  a screen  of  thickness  /,  be  I\  and  let  the  distances 
from  the  photometer  screen  be  pi  and  q±  when  the  latter  is 
equally  illuminated  by  both  sources.  Then 

h'  = (pi2+qi2)I" 

and  since  the  intensity  of  P before  the  screen  was  inserted 
was  /'  = (p2-r-q2)I",  we  have  for  the  transmission  coefficient 
for  the  thickness  t, 

r i pi2  q2 

a = — — = — — — 

r P 2 qx 2 

45.  THE  PHOTOMETER. 

To  study  the  relation  between  intensity  of  illumination 
and  distance  from  source,  and  to  determine  the  candle  power 
of  an  incandescent  lamp  and  the  transmission  coefficient  of 
tracing  cloth. 

Apparatus. — Photometer  bench  with  attached  scale  and 
three  sliding  carriages,  three  incandescent  lamps  with  leads 
and  metal  covers,  photometer  screen,  two  pieces  of  tracing 
cloth  for  transmission  screens,  and  dark  chamber  to  cover 
the  whole  apparatus. 


240 


Manual  of  Experiments 


Description. — The  photometer  bench  consists  of  two 
heavy  metal  bars  over  two  meters  long,  on  supporting  legs. 
Three  metal  .carriages  are  free  to  slide  along  this  bench,  the 
middle  one  carrying  the  photometer  screen,  the  other  two 
bearing  the  sources  of  light  to  be  compared.  A two  meter 
scale  is  fastened  to  the  bench,  and  pointers  on  the  carriages 
indicate  their  relative  positions  on  this  scale.  The  apparatus 
is  screened  from  extraneous  sources  of  light  by  a large  dark 
chamber,  painted  dull  black  inside,  which  completely  covers 
it  except  for  a small  opening  in  the  cloth  curtain  front 
through  which  the  photometer  screen  may  be  observed. 

Directions.— Relation  of  intensity  of  illumination  to  dis- 
tance.— Place  the  two  incandescent  lamps  of  known  candle 
power,  P and  Q,  at  opposite  ends  of  the  bench  200  cm.  apart 
and  move  the  Bunsen  screen  to  such  a position  between  them 
that  the  grease  spot  appears  to  have  the  same  intensity  of 
illumination  as  the  rest  of  the  screen.  There  will  always  be 
some  difference  in  color  between  the  spot  and  the  rest  of  the 
screen,  but  both  sides  of  the  screen  should  appear  alike  when 
viewed  in  the  mirrors.  Use  one  eye  so  far  as  possible  in 
judging  this  equality  of  intensities.  The  candle  powers 
marked  on  the  lamps  represent  the  intensity  in  the  direction 
of  the  indicator  mark  on  the  lamp.  The  lamps  must  there- 
fore be  placed  so  that  this  indicator  mark  faces  the  photo- 
meter screen.  Note  the  distances  of  P and  Q from  the 
screen,  and  repeat  with  distances  150  cm.  and  100  cm. 
between  P and  Q.  Make  three  trials  in  each  case  and  use 
the  mean  value  for  the  position  of  the  screen.  With  your 
data  show  that  the  intensity  of  illumination  varies  inversely 
as  the  square  of  the  distance  from  the  source. 

Transmission  coefficient. — Screen  the  lamp  P by  placing 
a sheet  of  tracing  cloth  between  it£and  the  photometer 


In  General  Physics 


241 


srceen.  Then  with  the  lamps  150  cm.  apart,  move  the  photo- 
meter screen  to  the  position  of  equal  illumination  and  note 
distances  pi  and  qi.  Repeat  using  two  thicknesses  of  tracing 
cloth.  Using  the  values  p and  q for  the  same  distance  be- 
tween P and  Q when  P was  not  screened,  calculate  the  trans- 
mission coefficient  of  the  cloth  and  show  how  the  intensity 
of  illumination  varies  on  transmission  through  more  than 
one  thickness. 

Candle  power  of  an  incandescent  lamp. — Replace  the  lamp 
P by  the  third  lamp,  of  unknown  candle  power,  with  its  in- 
dicator mark  facing  the  screen,  and  move  the  photometer 
screen  to  the  position  of  equal  illumination.  Note  the  dis- 
tances p and  g,  and  determine  the  candle  power  of  this  lamp 
in  terms  of  that  of  Q. 

Problems.— 1.— A standard  candle  and  a gas  flame  are 
placed  a{  = 10)  feet  apart,  the  gas  flame  being  of  £(  = 6) 
candle  power.  Where  would  a photometer  screen  have  to 
be  placed  between  them  in  order  to  make  the  grease  spot 
disappear? 

2.  — If  the  illumination  at  a point  due  to  a certain  source 
is  a = ( 6)  candle  feet,  what  will  it  be  when  a screen  b(  = 3.) 
mm.  thick  with  a transmission  coefficient  of  c=(0.6)  per 
mm.  is  placed  between  the  source  and  the  point? 

3.  — A light  of  intensity  a(  = 32)  c.  p.  behind  a sheet  of 
glass  which  has  a transmission  coefficient  b(  = 0.5)  per  m.m. 
and  a second  light  of  c(  = 18)  c.  p.  at  a distance  d(  = 5) 
meters,  produce  equal  illumination  on  a Bunsen  screen 

placed  between  them,  and  at  a distance  e(  = 2)  meters  from 

* _ 

the  first  light.  What  is  the  thickness  of  the  glass? 

4.  — A lamp  of  a(  = 16)  c.  p.  is  to  be  compared  with  a 
standard  of  £(  = 8)  c.  p .when  the  distance  between  them  is 


242 


Manual  of  Experiments 


c(  = 2.)  meters.  If  the  Bunsen  screen  can  be  set  for  equality 
of  illumination  with  an  accuracy  ±d(  = 5.)  m.  m.,  what  is 
the  % error  in  the  determination  of  the  c.  p.? 


REFLECTION. 

Theory  and  Definitions. — D.  620,  642-652;  G.  522- 
525,  530,  534-541;  K.  815-828;  W.  327-332,  337,  338.) 

The  simple  laws  of  reflection  may  be  stated  as  follows: 

(1)  When  a ray  of  light  is  reflected  from  a- smooth  sur- 
face, the  incident  ray  and  the  reflected  ray  make  equal  angles 
with  the  normal  to  the  surface  and  lie  on  opposite  sides  of  it. 

(2)  The  reflected  ray  lies  in  the  plane  of  incidence , i.  e ., 
in  the  plane  through  the  incident  ray  and  the  normal. 

By-  ray”  we  mean  here  a line  of  propagation  of  a light 
wave. 

Wheu  the  rays  of  light  coming  from  a point  source  are 
reflected,  they  may  intersect  each  other  at  some  point  in 
front  of  the  mirror  before  entering  the  eye,  or  they  may  di- 
verge so  that  their  prolongations  will  intersect  at  some  point 
behind  the  mirror.  In  either  case  the  eye  preceives  the 
image  of  the  source  at  the  point  of  intersection.  In  the  first 
case  the  image  is  real  and  can  be  projected  on  a screen.  In 
the  second  case  it  is  called  a virtual  image  and  cannot  be  pro- 
jected on  a screen.  Each  point  of  a luminous  object  may 
be  considered  a point  source  and  the  images  of  these  sources 
will  together  form  an  image  of  the  object. 

In  order  that  we  may  express  the  relation  between  image 
and  object  distances  from  the  mirror  and  its  radius  by  a 
single  equation,  for  all  cases  of  plane  and  spherical  mirrors, 
it  is  necessary  to  adopt  certain  conventions.  We  will  call 
the  center  of  the  mirror,  its  pole , and  the  normal  to  the  mirror 


In  General  Physics 


243 


at  the  pole,  or  the  line  through  the  pole  and  the  center  of 
curvature,  the  axis  ol  the  mirror. 

(1)  All  distances  are  to  be  measured  along  the  axis  from 
the  pole  of  the  mirror . 

(2)  Distances  measured  from  the  pole  in  the  direction  in 
which  the  incident  light  travels  are  negative , those  measured  in 
the  opposite  direction  are  positive. 

According  to  these  conventions,  the  radius  of  curvature 
of  a plane  or  of  a concave  spherical  mirror  is  positive  and  that 
of  a convex  spherical  mirror  negative;  the  distance  of  a real 
image  is  positive,  that  of  a virtual  image  negative. 

Calling  the  radius  of  curvature  of  the  mirror  the 

distance  of  the  source  (or  luminous  object)  from  the  pole 
“u”  and  the  distance  of  the  image  from  the  pole  l6vfJ  it 
may  be  shown  that  the  relation  between  them  for  plane  and 
spherical  mirrors  is 

112 

— — (1) 

u v r 

In  the  case  of  spherical  mirrors  this  .relation  holds  true 
only  for  small  apertures , i.  e .,  when  the  diameter  of  the  cir- 
cular boundry  of  the  mirror  is  small  compared  with  its  radius 
of  curvature. 

It  is  easily  seen  that  a point  source  on  the  axis  of  the 
mirror  will  have  its  image  formed  on  the  axis  or  on  the  pro- 
longation of  the  axis  behind  the  mirror. 

From  equation  (1)  it  is  evident  that  in  the  case  of  a 
plane  mirror  where  the  radius  of  curvature  r is  infinitely 
great,  we  have 

l/u-\-l/v  — 0 , or  v=—u 

That  is,  the  image  formed  by  a plane  mirror  is  as  far 
behind  the  mirror  as  the  object  is  in  front  of  it,  and  the  image 


244 


Manual  of  Experiments 


is  always  virtual.  Since  in  this  case  the  normals  to  the  dif- 
ferent points  of  the  mirror  are  all  parallel,  every  point  of 
the  source  may  be  considered  as  lying  on  an  axis,  and  the 
image  of  each  point  will  be  on  the  prolongation  of  its  particu- 
lar axis.  As  a result  the  image  will  be  perverted;  that  is,  the 
right  hand  side  of  the  object  or  source  will  appear  as  the  left 
hand  side  of  the  image. 

Since  u and  v are  equally  involved  in  equation  (1)  it  is 
evident  that  in  the  case  of  real  images,  at  least,  the  source 
and  image  distances  are  interchangeable;  that  is,  a source 
at  a distance  u has  its  image  formed  by  a mirror  of  radius 
r at  a distance  v,  where  u,  v end  r satisfy  equation  (1) ; if  the 
source  be  placed  at  a distance  v its  image  will  be  formed  at 
a distance  u.  Two  points  so  situated  are  called  conjugate  foci. 

If  the  source  be  placed  at  the  center  of  curvature,  u = r 
and  we  have 

l/r+l/&=  1/r,  or  v — r 

That  is,  the  center  of  curvature  is  its  own  conjugate  focus. 

When  parallel  incident  rays  are  reflected  from  a spherical 
mirror,  the  point  of  intersection  of  the  reflected  rays  is  called 
the  principal  focus  of  the  mirror,  and  the  distance  of  this 
point  from  the  pole  is  the  focal  length , /,  of  the  mirror.  The 
focus  (or  source)  of  parallel  rays  is  at  an  infinite  distance, 
and  putting  u = infinity  and  v=f  in  equation  (1)  we  have 

l/f=2fr,  or  f=rf 2 

That  is,  the  focal  length  of  the  spherical  mirror  is  half 
its  radius  of  curvature.  For  a concave  mirror  f is  positive, 
for  a convex  mirror  negative. 

Replacing  2/r  by  its  equivalent  1//,  equation  (1)  becomes 


1 1 1 


In  General  Physics 


245 


If  we  substitute  different  values  of  u in  this  equation  we  get 
the  following  relations  in  the  case  of  a concave  mirror : 

for  u = infinity,  v — f,  for  u > r,  v < r, 

for  u = r,  v = r,  for  u < r,  v > r, 

for  u = /,  v = infinity,  for  u < f,  v = negative. 

Positive  values  of  p and  of  / correspond  to  real  images, 
negative  values  to  virtual  images,  hence  the  image  formed 
by  a concave  mirror  is  real , except  for  values  of  u less  than 
f,  in  which  case  it  is  virtual.  We  see  that  as  the  source  moves 
up  from  an  infinite  distance  to  the  principal  focus,  the  image 
moves  off  from  the  focus  to  an  infinite  distance,  the  two  points 
meeting  at  the  center  of  curvature. 

In  the  case  of  the  convex  mirror,  the  value  of  v is  always 
less  than  / and  negative  for  positive  values  of  u.  If  however 
the  rays  of  light  incident  on  the  mirror  converge  toward  a 
point  — u behind  the  mirror,  v may  become  positive,  and  the 
image  will  be  real. 

Since  the  object  and  its  image  subtend  equal  angles  at 
'the  mirror,  it  is  evident  that  the  relative  sizes  of  both  are 
in  the  ratio  of  their  respective  distances  from  the  mirror  or 
if  0 and  / represent  their  corresponding  dimensions, 

0 u 


The  apparent  position  of  an  image  is  sometimes  obtained 
by  the  parallax  method.  By  parallax  we  mean  the  apparent 
displacement  of  an  object  {or  image)  relatively  to  another  object , 
due  to  a real  displacement  of  the  observer.  If  the  two  objects 
are  coincident  or  are  equally  distant,  their  relative  parallax 
vanishes. 

When  waves  from  a point  source  fall  upon  a spherical 
mirror  obliquely,  they  do  not  form  a point  image,  but  there 


246 


Manual  of  Experiments 


will  be  formed  two  line  images  at  different  distances  and  at 
right  angles  to  each  other.  The  nearer  the  source  is  to  the 
axis  of  the  mirror,  the  more  nearly  these  focal  lines  will 
coincide,  and  will  be  reduced  to  form  a point  image  on  the 
axis. 

When  the  source  of  light  is  not  a point,  but  an  extended 
object,  each  point  of  this  object  will  have  its  own  point 
image  formed,  and  all  of  these  images  together  make  up 
the  image  of  the  source.  But  those  points  which  lie  off 
of  the  axis  of  the  mirror  will  have  their  images  formed  at 
different  distances  from  the  pole  than  that  of  the  point  on 
the  axis.  It  will  therefore  be  impossible  to  find  a position 
of  the  screen  at  which  all  parts  of  the  projected  image  are 
equally  in  focus. 

46.  MIRRORS. 

To  show  that  the  angle  of  incidence  is  equal  to  the  angle 
of  reflection,  and  to  find  the  position,  size  and  shape  of  the 
image  formed  by  a plane  mirror;  to  show  the  relation  between 
image  and  object  distance  and  focal  length  of  a concave 
spherical  mirror,  and  to  measure  its  focal  length. 

Apparatus. — Optical  bench,  incandescent  lamp  with 
cover  and  slit,  plane  and  concave  spherical  mirrors  and  hold- 
ers, sliding  blocks,  right  triangle,  protractor,  pins,  screen 
with  slit  and  cross  wires,  and  sheet  of  paper. 

Description. — The  optical  bench  is  a slotted  channel 
iron  bench  about  one  and  a half  meter  long,  with  a mm. 
scale  attached  to  one  side.  (See  Fig.  63.)  The  mirrors, 
lenses  and  screens  to  be  experimented  with  are  carried  by 
metal  blocks  which  are  arranged  to  slide  along  the  slot. 


In  General  Physics 


247 


Their  positions  on  the  scale  may  be  read  by  means  of  an 
indicator  mark  on  the  side  of  each  block.  One  block  carries 
a screen  with  one  side  enameled  white  so  that  images  pro- 
jected on  it  may  be  sharply  focussed.  It  has  also  a rectangu- 
lar opening  in  the  center,  across  which  cross  wires  are 
stretched.  These  may  be  used  as  the  object  to  be  focussed 
on  the  screen  when  a strong  light  is  placed  behind  the  open- 
ing. An  incandescent  lamp  is  covered  with  a copper  screen, 
in  one  side  of  which  is  a rectangular  slit.  This  generally 
serves  as  the  object  to  be  projected  on  the  white  screen,  and 
it  is  this  opening,  not  the  incandescent  filament  of  the  lamp, 
which  must  be  sharply  focussed.  To  aid  in  focussing,  a 
sharp  spur  is  cut  on  one  side  of  the  slit.  The  mirrors  and 
lenses  are  mounted  so  that  they  may  be  turned  about  both 
a vertical  and  a horizontal  axis,  in  order  to  place  them  in 
any  desired  position,  and  the  indicator  mark  on  the  carrying 
block  is  in  all  positions  under  the  pole  of  the  mirror  or  lens. 
On  the  front  of  these  blocks  are  a pair  of  clamps  for  holding 
a small  plane  mirror  for  use  in  the  parallax  method  of  fo- 
cussing. 

Directions. — Relation  between  angles  of  incidence  and 
reflection. — The  plane  mirror  is  fastened  to  one  face  of  a 
supporting  block.  Draw  a straight  line  across  the  sheet  of 
paper  and  place  the  silvered  (reflecting)  side  of  the  mirror 
along  this  line.  Stick  a black  headed  pin  about  10  cm.  in 
front  of  the  mirror  and  a second  ordinary  pin  to  one  side  of 
the  former  and  about  2 cm.  in  front  of  the  mirror.  Move 
the  eye  on  a level  with  the  mirror  until  a position  is  found 
where  the  images  of  the  two  pins  coincide.  Place  a third 

pin  c in  line  with  the  coincident  images  and  the  eye.  Then 

■ 

draw  the  line  through  a and  b to  the  reflecting  surface  and 
a perpendicular  to  this  surface  at  the  point  of  intersection  o. 
Also  join  c and  o.  See  Fig.  61.  If  the  glass  of  the  mirror 


248 


Manual  of  Experiments 


is  thin  the  deviation  of  the  rays  ao  and  co,  by  refraction, 
may  be  neglected.  Measure  the  angle  of  incidence  and  the 
angle  of  reflection  and  compare  their  values.  Place  the  pins 
a and  b in  different  positions  and  repeat,  using  large  angles 
for  the  sake  of  accuracy. 


Fig.  61 


Position , size  and  shape  of  the  image  in  a plane  mirror. — 
Draw  a line  on  the  other  side  of  the  paper,  through  its  center 
and  place  the  mirror  as  before.  Lay  the  triangle  in  front 
of  the  mirror,  trace  its  outline  with  a sharp  pencil,  and  mark 
the  angles  a , b and  c.  Place  a short  pin  at  the  vertex  a and 
a large  pin  a'  behind  the  mirror,  a'  is  to  be  placed  at  the 
image  of  a by  the  method  of  parallax,  hence  a'  must  be 
moved  to  different  positions  until  such  a position  is  reached 
that  upon  turning  the  head  from  side  to  side  or  looking  at 
the  mirror  from  different  positions,  the  image  of  a in  the 
mirror  coincides  with  the  pin  a'  as  seen  above  the  mirror, 
for  all  angles  of  incidence.  So  long  as  a'  is  not  at  the  image 
of  a there  will  appear  to  be  relative  motion  between  these 
two  as  the  head  is  turned  from  side  to  side.  Locate  the 
images  V and  c'  of  b and  c respectively  in  the  same  way. 
Join  the  points  a,  b and  c to  their  corresponding  images. 
Measure  and  compare  the  perpendicular  distance  of  each 
vertex  from  the  mirror,  with  that  of  its  image.  Join  a\  b' 
and  c'  to  form  the  image  of  the  triangle.  Measure  and  com- 
pare the  lengths  of  its  sides  with  that  of  the  corresponding 
sides  of  the  triangle. 


In  General  Physics 


249 


Focal  length  of  a concave  spherical  mirror. — Place  the 
mirror  on  the  optical  bench  facing  the  screen;  carry  them  to 
a shaded  window,  and  turn  bench  and  mirror  toward  some 
distant  object,  such  as  a church  tower,  or  a tree.  Move  the 
screen  on  the  bench  until  a sharply  outlined  image  of  this 
object  is  obtained.  Have  the  angles  of  incidence  and  re- 
flection as  small  as  possible.  This  image  is  formed  at  the 
principal  focus.  Measure  the  focal  length  between  screen 
and  pole  of  mirror. 

Relation  between  distances  of  object  and  of  real  images  from 
a spherical  mirror. — Return  the  apparatus  to  the  place  as- 
signed for  performing  the  experiment.  Arrange  the  lamp 
close  to  the  bench  so  that  the  light  passing  through  the  slit 
may  be  reflected  from  the  mirror  to  the  screen,  the  slit  being 
on  the  same  level  with  the  mirror  and  screen.  To  get 
sharper,  though  less  brilliant  images,  the  mirror  may  be 
covered  by  a diaphram  made  of  paper  or  sheet  metal,  having 
the  outline  of  the  mirrox  and  a circular  hole  at  the  center  one 
to  one  and  a half  inches  in  diameter.  The  rectangular  slit 
is  the  object  whose  image  is  to  be  formed  and  to  aid  in  getting 
the  image  sharp  a fine  spur  is  cut  so  that  it  projects  at  one 
side  of  the  slit.  The  distances  u and  v,  of  the  slit  and  of 
the  screen  respectively  from  the  center  of  the  mirror  may 
be  measured  on  the  scale.  It  should  be  remembered  that 
for  oblique  incidence  (which  is  unavoidable  here)  two  images 
are  formed  at  slightly  different  distances  and  a mean  position 
must  be  found  in  each  case. 

Place  the  slit  close  to  and  in  the  same  plane  with  the 
screen  and  move  the  mirror  until  a sharp  image  of  the  slit 
(not  the  lamp  filament)  appears  on  the  screen.  The  dis- 
tance between  mirror  and  screen  is  now  the  radius  of  curva- 
ture of  the  latter.  Measure  this  distance  r and  compare  it 
with  the  focal  length  /. 


250 


Manual  of  Experiments 


This  distance  may  also  be  determined  by  removing  the 
disk  with  the  slit  from  the  lamp  cover,  placing  the  lamp  di- 
rectly behind  the  white  screen,  at  one  end  of  the  bench  and 
adjusting  the  position  of  the.  mirror  until  a sharp  image  of 
the  cross  wires  is  projected  on  the  screen  just  beside  the 
slit.  The  object  and  image  are  then  both  at  the  distance 
r from  the  mirror. 

With  the  lamp  cover  and  slit  once  more  at  the  side  of 
the  bench,  place  the  mirror  at  a distance  between  / and  r, 
from  the  slit  and  move  the  screen  until  a sharp  image  of 
the  slit  is  formed.  Measure  the  distances  u and  v on  the 
scale,  and  the  length  of  the  slit  and  of  its  image  with  the  aid 
of  dividers.  Repeat  with  the  mirror  at  a distance  greater 
than  r from  the  slit.  Compare  the  sum  of  the  reciprocals 
of  u and  v in  the  two  cases  with  each  other  and  with  the  re- 
ciprocal of  the  focal  length  /.  Also  compare  the  ratio  of 
the  object  and  image  lengths  in  each  case  with  the  corre- 
sponding ratio  of  u and  v. 

Virtual  image. — Remove  the  diaphram  and  fasten  a short 
pin  in  the  sliding  block  at  a distance  from  the  mirror  less 
than  the  focal  length,  f.  With  a large  pin  behind  the  mirror 
locate  the  image  of  the  short  pin  by  the  method  of  parallax, 
as  described  for  a plane  mirror. 

Measure  the  distances  u and  v and,  remembering  the 
adopted  convention  concerning  signs,  compare  the  sum  of 
the  reciprocals  of  u and  v with  the  reciprocal  of  f. 

Owing  to  the  spherical  aberration,  the  image  of  the  pin 
at  the  edges  of  the  mirror  curves  from  side  to  side  as  the  eye 
is  moved,  and  makes  it  difficult  to  determine  these  conjugate 
foci.  When  this  curvature  is  objectionable  the  following 
method  should  be  used:  Place  a small  plane  mirror  M in 

front  of  the  lower  half  of  the  spherical  mirror,  C.  (See  Fig. 
62.)  Place  a black  headed  pin  0 at  a distance  less^fchan  f 


In  General  Physics 


251 


in  front  of  C , and  a white  headed  pin  P in  front  of  M.  By 
looking  down  at  an  angle  at  both'  mirrors,  the  image  of 
0 in  C,  (/),  and  of  P in  Af,  (0,  may  be  seen  simultaneously 
without  interference  from  the  pins  themselves.  P must 
now  be  shifted  in  position  until  there  is  no  parallax  between 
the  images  / and  <2,  when  the  eye  of  the  observer  moves 


from  side  to  side.  I and  Q then  occupy  the  same  position, 
and  Q is  as  far  behind  the  reflecting  surface  of  M as  P is  in 
front  of  it.  Calling  this  distance  “ui”  and  the  distance 
between  reflecting  surfaces  of  C and  M “ d,”  we  see  that  the 
distance  of  I from  C is 

v = Ui~d 

Note  on  the  scale  the  distances  u,  d and  calculate  v,  and 
compare  the  reciprocals  of  f with  the  sum  of  the  reciprocals 
of  u and  v. 

Problems. — 1. — Draw  large  clear  diagrams  showing  the 
formation  of  a real,  and  of  a virtual  image  of  an  arrow,  by 


252  Manual  of  Experiments 

means  of  a concave  mirror,  when  the  arrow  is  perpendicular 
to  and  symmetrical  to  its  axis.  Trace  the  path  of  two  rays 
from  each  end  of  the  arrow;  one  parallel  to  the  axis  of  the 
mirror,  or  one  through  its  center  of  curvature,  and  one 
through  its  principal  focus.  Mark  center  C,  focus  F,  object 
(9,  image  /,  and  the  direction  of  the  rays  with  arrow  heads. 
Use  full  lines  for  actual  rays,  dotted  lines  for  construction 
and  prolongation  of  rays. 

2.  — In  the  same  way  draw  diagram  of  image  formation 
by  a convex  spherical  mirror. 

3. - — a ) A concave  spherical  mirror  has  a radius  of 
curvature  of  #(  = 30)  cm.  Find  the  position  and  size  of  the 
image  formed  when  a candle  flame  with  a height  b(  = 2) 
cm.  is  placed  r(  = 20)cm.  in  front  of  the  mirror,  on  its  axis. 
b)  Find  the  position  and  size  of  the  image  when  the  candle 
is  moved  a distance  J(  = 8)  cm.  toward  the  mirror. 

4.  — A spherical  mirror  forms  an  image  of  an  object 
having  a height  #(  = 5)  inches,  and  placed  b(  = 20)  inches 
in  front  of  the  mirror,  a)  If  the  image  is  at  a distance 
c(  = 12)  inches  behind  the  mirror,  find  the  focal  length  of 
the  latter,  b)  Is  the  mirror  concave  or  convex?  c)  What 
is  the  height  of  the  image? 

REFRACTION  THROUGH  LENSES. 

Theory  and  Definitions. — (D.  620,  656,  666-667;  G. 
552,  562-572;  K.  834-836,  849-858;  W.  348,  349.)  A lens  is 
a portion  of  a refracting  medium  bounded  by  two  curved 
surfaces,  which  are  usually  spherical.  A line  through  the 
centers  of  curvature  of  the  two  surfaces  is  termed  the  prin- 
cipal axis  of  the  lens.  The  points  where  this  axi£\quts  the 
two  surfaces  are  termed  the  poles  of  the  surfaces.  In  the 


In  General  Physics 


253 


following  discussion  the  lenses  are  supposed  to  be  thin;  i.  e., 
the  two  poles  are  so  close  together  that  distances  may  be 
measured  from  the  center  of  the  lens.  For  definitions  of 
aperture,  real  and  virtual  images,  see  previous  theory  of 
reflection. 

Lenses  are  of  two  general  kinds.  Those  which  are  thicker 
at  the  poles  than  at  the  periphery  usually  cause  the  incident 
light  from  a point  source  to  converge  after  refraction  to  a 
point  beyond  the  lens,  when  surrounded  by  a less  refractive 
medium,  and  are  called  convergent.  Those  which  are  thinner 
at  the  poles  than  at  the  periphery  cause  the  incident  light 
to  diverge  after  refraction  from  a point  in  front  of  the  lens, 
when  surrounded  by  a less  refractive  medium,  and  are  called 
divergent  lenses.  When  the  incident  light  is  parallel  to  the 
principal  axis  the  lens  will  cause  it  to  converge  to,  or  diverge 
from  a point  called  the  principal  focus.  The  distance  from 
the  lens  to  the  principal  focus  is  called  the  focal  length  of 
the  lens.  The  focal  length  remains  unaltered  when  the  lens 
is  reversed. 

We  shall  adopt  the  same  conventions  for  signs  as  in  the 
case  of  reflection;  namely: — 

(1)  All  distances  are  to  be  measured  from  the  poles  along 
the  axis. 

(2)  Distances  measured  from  the  poles  in  the  direction  in 
which  the  incident  light  travels  are  negative , those  measured  in 
the  opposite  direction  are  positive. 

According  to  this  convention  the  focal  length  of  a con- 
vergent lens  is  negative,  that  of  a divergent  lens  positive. 

Let  u be  the  distance  of  an  object  from  the  lens  center, 
v the  distance  of  its  image?  and  / the  focal  length.  Then 
for  lenses  having  a small  aperture , the  following  relation 
holds  for  both  kinds  of  lenses: — 


254 


Manual  of  Experiments 


1 i i 

U V f 

It  should  be  remembered  that  besides  the  signs  of  the 
fractions  in  this  equation  each  symbol  (u,  v,  /,)  has  its  own 
sign  depending  upon  the  conventions  given  above. 

This  equation  gives  us  the  following  relations: — 


Convergent  lenses: 

For  u — infinity, 
For  oo  > u > 2/, 
For  2/  > u > /, 
For  / > u > o, 


v = —f. 
—f>v>  -2 f 

- 2/  > v > - o°. 
oo  > v > o. 


In  the  last  case  the  image  is  virtual.  The  other  images  are 
all  real. 


Divergent  lenses. — 

For  u = 00 , v — f. 

For  °o  > u > o,  + f > v > o. 

The  image  formed  by  the  divergent  lens  is  always  virtual 
and  always  nearer  to  the  lens  than  the  object. 

Since  the  object  and  image  subtend  equal  angles  at  the 
center  of  the  lens,  their  sizes  will  be  in  the  same  ratio  as 
their  respective  distances  from  the  lens.  Hence,  if  0 and 
I represent  their  corresponding  dimensions 

0/I  = u/v. 

If  a lens  forms  an  image  of  an  object  on  a screen  when 
it  is  half  way  between  them,  it  may  be  shown  that  the  dis- 
tance between  them  is  four  times  its  focal  length,  f.  If  the 
fixed  distance  between  object  and  screen  is  greater  than  4/, 
there  will  be  two  positions  of  the  lens  between  them,  at  which 


In  General  Physics 


255 


it  forms  a distinct  image  of  the  object  on  the  screen.  This 
fact  suggests  a method  of  determining  the  focal  length  of 
the  lens.  For  if  s be  the  distance  between  the  object  and 
the  screen  (greater  than  4/),  and  / be  the  distance  between 
the  two  positions  of  the  lens,  at  which  it  projects  a sharp 
image  on  the  screen,  it  may  be  shown  that  the  focal  length  is 


s2-l2 


47.  LENSES. 

To  study  the  relation  between  focal  length,  image  and 
object  distance,  and  image  and  object  size  in  the  case  of 
convex  and  concave  lenses,  and  to  measure  their  focal  lengths. 

Apparatus. — Optical  bench,  double  convex  and  double 
concave  lens  and  stand,  incandescent  lamp  and  leads,  cover 
and  slit,  sliding  screen,  two  sliding  blocks  meter  stick  and 
pins.  (Apparatus  described  in  Exp.  46.) 

Directions. — Convex  lens. — Take  the  optical  bench  to 
a shaded  window.  Place  the  convex  lens  in  its  holder  at 
one  end  of  the  bench  with  its  axis  pointed  toward  some 
distant  object,  such  as  a chimney  or  church  steeple.  Move 
the  screen  along  the  bench  until  a sharp  image  of  the  object 
is  projected  on  it.  The  rays  from  the  object  are  practically 
parallel  and  the  distance  from  lens  to  screen  is  the  focal 
length  of  the  former.  Measure  this  length  on  the  attached 
scale. 

Place  the  lamp  with  its  slit  vertically  over  zero  of  the 
scale  and  place  the  lens  at  a distance  between  / and  2 f from 
the  slit.  Fig.  63.  Then  move  the  screen  until  there  is  a 
sharp  image  of  the  slit  (not  of  the  lamp  filament)  upon  it. 


256 


Manual  of  Experiments 


Adjust  for  sharp  definition  by  observing  the  spur  on  the  side 
of  the  slit.  Measure  the  distances  u and  v and  also  the 
lengths  0 and  / of  the  object  and  image.  Repeat  with  the 


lens  at  a distance  equal  to  2/  and  then  at  a greater  distance 
from  the  slit.  Compare  —l/u  + l/v  with  1 If  in  both  cases, 
remembering  the  adopted  conventions  regarding  signs. 
Also  compare  O/I  with  u/v. 

Virtual  image. — Place  a block  with  a small  pin  behind  the 
lens  at  less  than  the  focal  distance  f.  Locate  the  virtual 
image  of  this  pin  by  the  method  of  parallax;  i.  <?.,  place  a 
second  long  pin  on  a sliding  block  behind  the  first  pin  and 
move  the  second  on  the  bench  until  it  is  at  such  a position 
that  the  image  of  the  first  pin  seen  through  the  lens,  and  the 
top  of  the  second  pin  seen  above  the  lens  coincide  for  all 
positions  of  the  eye  in  front  of  the  lens.  (The  second  pin 
is  then  at  the  virtual  image  of  the  first  pin.)  See  Fig. 
64.  Measure  the  distances  u and  v and  compare  — l/u  + l/v 
with  1//. 

If  the  curvature  of  the  image  at  the  edge  of  the  lens 
due  to  spherical  aberration  is  appreciable,  it  is  better  to 
place  a small  plane  mirror  in  front  of  the  lens,  and  proceed  as 
described  in  Experiment  46,  to  find  the  virtual  image.  The 
only  change  in  this  method  is  that  the  pin  OTsjm  the  same 
side  of  the  lens  with  the  two  images  I and  <9,  and  its  image 
is  seen  through  the  lens. 


In  General  Physics 


257 


The  distances  u,  d and  uh  are  to  be  measured  and  v cal- 
culated, in  order  to  substitute  u and  v in  the  lens  equation. 


Concave  lens. — The  images  are  all  virtual  and  are  located 
by  the  parallax  method.  Take  the  optical  bench  to  a shaded 
window  and  locate  the  image  of  a distant  vertical  line  such 
as  the  corner  of  a building,  by  sliding  a long  pin  along  the 
bench,  behind  the  lens,  until  it  coincides  with  the  image 
of  the  object  for  all  angles  of  observation.  Measure  / from  * 
the  lens  to  the  pin.  As  in  the  previous  case  of  a virtual 
image,  so  here  also,  it  is  preferable  to  use  the  parallax  method 
described  in  Experiment  46. 

Place  a second  pin  at  a distance  2 f from  the  lens  and  lo- 
cate its  image.  Measure  u and  v.  Repeat  with  the  pin  at  a 
distance  / from  the  lens.  Compare  — 1/u  + l/v  with  1/f. 

Problems. — 1. — Draw  large,  clear  diagrams  showing' 
the  formation  of  a real  and  of  a virtual  image  of  an  arrow 
by  means  of  a double  convex  lens,  the  arrow  being  perpen- 
dicular to  and  symmetrical  to  the  principal  axis.  Trace  the 
path  of  two  rays  from  each  end  of  the  arrow;  one  parallel 
to  the  principal  axis,  and  one  through  the  principal  focus,  or 
one  through  the  center  of  the  lens.  Mark  object  (9,  image  /, 


258  Manual  of  Experiments 

principal  focus  F,  and  direction  of  rays  by  means  of  arrow 
heads.  Use  full  lines  for  actual  rays,  dotted  lines  for  con- 
struction and  prolongation  of  rays. 

2.  — Lay  off  on  coordinate  paper  a pair  of  rectangular 
axes  intersecting  in  the  upper  left  hand  corner.  On  these 
axes  lay  off  the  corresponding  values  of  u and  v obtained 
above  for  the  real  images  of  a convex  lens  as  abscissae  and 
ordinates  respectively.  Connect  corresponding  values  of 
u and  v by  straight  lines.  Compare  the  ordinates  of  the 
common  point  of  intersection  of  these  lines,  with  each  other 
and  with  the  focal  length  of  the  lens  as  determined  exper- 
imentally  above.  Using  the  lens  equation,  show  by  the 
proportion  between  the  sides  of  the  similar  triangles  formed 
by  one  of  the  oblique  lines,  that  the  ordinate  of  this  point 
should  equal  the  focal  length  of  the  lens. 

3.  — An  object  which  is  a(  = 20)  inches  high  is  looked  at 
through  a double  concave  lens  of  b(  = 15)  inches  focal 
length  when  it  is  c(  = 30)  inches  from  the  lens.  Find  the 
size  and  position  of  the  image  formed. 

4.  — A double  convex  lens  of  a{  = 8.6)  cm.  focal  length 
casts  a sharp  image  of  a lamp  on  the  screen,  when  placed 
between  them  at  a certain  point.  When  it  is  moved  a 
distance  b(  = 15)  cm.  nearer  to  the  screen  it  again  forms  a 
sharp  image.  Find  a)  the  distance  between  lamp  and  screen, 
and  b)  the  distance  from  lamp  to  lens  in  its  first  position. 

48.  INDEX  OF  REFRACTION  OF  A LENS. 

To  determine  the  index  of  refraction  of  a lens  from  its 
focal  length  and  the  radii  of  curvature  of  its  faces.  (D. 
668;  G.  539,  545;  K.  849-851;  W.  351,  352^ 

Apparatus. — Optical  bench,  double  convex  lens,  sphero- 
meter,  piece  of  plate  glass,  millimeter  scale. 


In  General  Physics 


259 


Description  and  Theory. — When  parallel  rays  of  light 
fall  on  a double  convex  lens  they  converge  to  a point  called 
the  principal  focus  of  the  lens.  The  reciprocal  of  the  dis- 
tance f,  from  the  center  of  the  lens  to  this  point  is  given  by 

1 1 1 

— = (ft  — 1)  ( ) (1) 

/ ri  r2 

(see  references  above  for  development)  where  r\  is  the  radius 
of  curvature  of  the  face  upon  which  the  light  impinges,  r2 
the  radius  of  curvature  of  the  other  face  and  n the  index  of 
refraction  of  the  lens. 

The  focal  length  of  the  lens  may  be  obtained  by  allowing 


rays  of  light  from  a distant  object  (tree,  building,  etc.)  to 
fall  on  the  lens  and  be  brought  to  a focus  on  a screen  placed 
behind  the  lens.  When  a clear  image'is  obtained,  the  dis- 


260 


Manual  of  Expekiments 


tance  from  the  screen  to  the  center  of  the  lens  will  give 
the  focal  length  f.  The  radii  of  curvature  of  the  two  faces 
of  the  lens  may  be  obtained  by  use  of  the  spherometer. 

The  spherometer  consists  of  a screw  of  one  millimeter 

•x 

pitch  which  turns  in  a nut  supported  by  an  equilateral 
tripod.  (See  Fig.  65).  The  axis  of  the  screw  passes 
through  the  center  of  the  circle  in  which  the  three  feet  of 
the  tripod  are  situated.  The  whole  number  of  turns  made 
by  the  screw  may  be  observed  from  the  vertical  millimeter 
scale  erected  over  one  of  the  legs  of  the  tripod.  l,  At  the  head 
of  the  screw  is  a circular  disk  which  is  divided  inter  100  equal 
parts.  Since  the  pitch  of  the  screw  is  one  millimeter,  one 
complete  turn  represents  a longitudinal  motion  of  the  screw 
of  one  millimeter  and  turning  the  screw  through  one  of  the 
small  divisions  on  the  disk  represents  a longitudinal  motion 
of  the  screw  of  one-hundredth  of  a millimeter.  If  the  plane 
of  the  disk  is  between  6 and  7 on  the  vertical  scale  and  the 
forty-fifth  division  of  the  disk  is  in  line  with  the  face  of  the 
vertical  scale,  the  reading  is  6.45  mm.  Tenths  of  a division 
on  the  disk  may  be  estimated. 

To  find  the  radius  of  curvature  of  a spherical  surface, 
it  is  first  necessary  to  determine  the  position  of  the  disk 
when  the  three  legs  and  the  point  of  the  screw  are  in  the  same 
plane.  This  is  done  by  placing  the  spherometer  upon  the 
glass  plate  and  turning  the  screw  down  till  its  point  rests 
upon  the  glass.  Then  raise  the  screw  and,  placing  the 
spherometer  on  the  convex  lens,  turn  the  screw  down  again 
until  it  comes  in  contact  with  the  glass  at  the  point  Fig. 
66.  The  difference  between  this  reading  and  the  “zero” 
reading  is  the  amount  the  point  of  the  screw  has  been  re- 
moved from  the  plane’ of  the  ends  of  the^three  legs.  Let 
this  distance  be  represented  by  a , the  distance  between  the 
legs  and  the  point  of  the  screw  (when  the  points  of  the  screw 


In  General  Physics 


261 


and  the  legs  are  in  the  same  plane)  by  b , and  the  radius  of 
curvature  by  R.  It  may  be  seen  from  Fig.  66  that 

R2  = (R-a)2+b2  (2) 


Solving  this  for  R gives 

a2-\-b2  a b 2 

R = — = 1 ....  (3) 

2 a 2 2 a 

Directions. — (1)  To  find  the  radii  of  curvature  of  the 
faces  of  the  lens,  first  place  the  spherometer  on  a sheet  of 
plate  glass  and  turn  the  screw  down  till  it  touches  the  plate. 
To  test  for  this  adjustment,  apply  a very  slight  pressure 
immediately  above  each  of  the  three  legs  and  test  to  see 


Fig.  66 


whether  the  spherometer  rocks  on  the  point  of  the  screw. 
Adjust  the  screw  until  the  position  is  found  where  this  mo-  ' 
tion  just  disappears.  Read  the  position  of  the  top  of  the 
disk  in  millimeters  by  means  of  the  vertical  scale  and  the  frac- 
tion of  a millimeter  by  means  of  the  divisions  on  the  circular 


262 


Manual  of  Experiments 


disk.  The  tenths  of  the  small  divisions  can  be  estimated 
by  the  eye.  Move  the  spherometer  to  other  positions  upon 
the  plate  and  make  10  readings  as  above,  for  the  “zero 
position”  of  the  screw.  Take  the  mean  of  the  10  readings 
for  the  true  value  of  the  ‘ 1 zero  position.  ’ ’ Raise  the  screw, 
place  the  spherometer  on  the  convex  lens  and  then  turn  the 
screw  down  till  the  point  makes  contact  in  the  same  way 
as  it  did  on  the  glass  plate.  Note  the  reading  as  indicated 
by  the  vertical  scale  and  the  divided  disk.  Move  the  sphero- 
meter to  different  positions,  make  10  different  settings  of  the 
screw  and  record  the  reading  for  each  setting.  Take  the 
mean  of  the  10  readings  and  find  the  difference  between 
it  and  the  mean  “zero  position.”  This  difference  is  the 
distance  a in  the  formula  above  and  is  in  millimeters.  Record 
the  difference  in  centimeters.  To  find  the  distance  b 
between  the  legs  and  the  point  of  the  screw,  place  the  sphero- 
meter upon  a piece  of  paper  on  the  plate  glass  and  make  the 
impression  of  the  points  of  the  legs  and  screw  upon  the  paper. 
With  a millimeter  scale  measure  the  distance  between  the 
central  impression  made  by  the  screw  and  the  inside  part  of 
the  impressions  made  by  the  legs.  The  inside  of  the  im- 
pressions made  by  the  legs  is  chosen  because,  the  surface 
being  convex,  the  inside  of  the  end  of  each  leg  is  the  part  that 
rests  upon  the  surface.  Make  two  such  sets  of  impressions 
and  find  the  mean  value  of  the  6 determinations  of  the  dis- 
tance from  the  central  impression  to  the  outer  ones  made  by 
the  legs.  Call  this  mean  value  b and  record  it  in  centi- 
meters. From  the  values  of  a and  b thus  obtained,  calculate 
the  value  of  the  radius  of  curvature  from  equation  (3). 

In  like  manner  find  the  radius  of  curvature  of  the  other 
face  of  the  lens. 

(2)  To  find  the  focal  length/,  place  thedens  in  the  holder 
on  the  optical  bench  and  adjust  the  screen  so  that  rays  from  a 


In  General  Physics 


263 


distant  object  come  to  a focus  on  it.  Measure  the  distance 
from  the  screen  to  the  center  of  the  lens  and  record  it  in 
centimeters.  Make  five  settings  of  the  screen  and  take  the 
average  as  the  value  of  /. 

Substitute  the  values  of  f,  tt  and  r2  in  equation  (1)  and 
solve  for  n , the  index  of  refraction  of  the  lens.  Care  must 

be  taken  to  assign  to  rx  and  r2  their  proper  algebraic  sign. 

Questions. — Draw  figure  and  derive  formula  for  radius 
of  curvature  R , of  a concave  surface.  What  distance  should 
be  taken  as  b ? Give  reason  for  your  answer.  If  an  error 

of  .5  mm  were  made  in  measuring  b , what  per  cent  error 

would  it  make  in  the  value  of  R in  your  results? 


ALTERNATIVE  METHOD 

When  a spherometer  is  not  available  for  measuring  the 
radii  of  curvature  of  the  lens,  the  following  method  may  be 
used: 

Theory  and  Method. — If  light  from  a source  0 falls 
on  a double  convex  lens  placed  so  that  the  light  is  reflected 
from  the  back  surface  of  the  lens  and  forms  an  image  of  the 
source  at  the  same  point  0,  it  is  evident  that  within  the  lens 
the  rays  are  perpendicular  to  the  back  surface,  otherwise  they 
would  not  retrace  their  incident  path.  This  image  is  in 
exactly  the  same  position  as  it  would  have  been  if  it  were 
formed  by  light  waves  falling  perpendicularly  upon  the  back 
of  the  lens  and  converging  toward  the  point  of  intersection  of 
the  projection  of  the  rays  within  the  lens,  namely  P.  This 
point  is  also  the  center  of  curvature  of  the  reflecting  surface. 
(See  Fig.  67.)  The  distance  PC  will  be  the  apparent  object 
distance  u of  such  waves,  and  OC  the  corresponding  image 


264 


Manual  of  Experiments 

distance  v.  If  / be  the  focal  length  of  the  lens  we  have 

1/ /=  — 1/u+l/v, 

and  as  v and  / are  both  measurable  quantities  we  may  cal- 
culate the  distance  PC  from  this  formula.  If  the  lens  is 
very  thin  this  distance  is  practically  the  radius  of  curvature 
r2  of  its  back  surface,  but  for  an  ordinary  lens  we  must  add 
to  this  one  half  the  thickness  of  the  lens  at  its  center. 

The  lens  is  then  reversed  and  the  radius  n of  its  other  face 
is  determined  in  the  same  way.  Knowing  the  focal  length 
and  the  radii  of  curvature  of  the  lens  faces,  the  index  of  re- 
fraction may  be  calculated  by  the  formula  already  given. 

Directions. — With  the  screen  and  cross  wires  at  one 
end  of  the  bench,  place  the  lamp  behind  the  screen  and  move 


In  General  Physics 


265 


the  lens  along  the  bench  in  front  of  it  until  a sharp  reflected 
image  of  the  wires  is  projected  on  the  screen.  By  turning 
the  lens  slightly  the  image  may  be  seen  just  beside  the  open- 
ing in  the  screen.  Note  the  distance  v between  the  cross 
wires  and  center  of  lens.  Then  reverse  the  lens  and  repeat 
the  measurement.  With  vernier  or  micrometer  calipers 
measure  the  thickness  of  the  lens  at  its  center,  taking  the 
mean  of  five  or  six  measurements.  From  these  data  calcu- 
late the  radii  /r  and  r2.  Next  determine  the  focal  length  of 
the  lens  by  one  of  the  methods  described  in  Experiment 
47.  Finally  calculate  the  index  of  refraction  by  formula  (1). 

Question. — What  % "error  would  result  from  neglecting 
the  thickness  of  the  lens  in  calculating  the  radii  of  curvature? 


49.  THE  CRITICAL  ANGLE  FOR  GLASS  IN  AIR 

To  determine  the  index  of  refraction  and  the  critical 
angle  for  glass  in  air.  (D.  656-678;  G.  547,  550;  K.  835, 
840,  841;  W.  343,  344.) 

Apparatus. — Glass  cube,  straight  edge,  long  pin,  and 
sheet  of  paper. 

Theory  and  Method. — When  a light  ray  passes  from 
one  transparent  medium  into  a denser  one  at  any  angle  of 
incidence  except  0°  or  90°  it  will  be  bent  toward  the  normal 
to  their  interface.  But  if  the  refraction  takes  place  from  the 
dense  into  the  rarer  medium,  the  ray  will  be  bent  away  from 
the  normal.  In  other  words  the  angle  of  refraction  increases 
more  rapidly  than  the  angle  of  incidence  as  we  increase  the 
latter  from  0°.  It  is  evident  that  the  angle  of  refraction 
can  not  exceed  90°,  hence  for  some  angle  of  incidence  less 


266 


Manual  of  Experiments 


than  90°  refraction  will  cease  and  the  incident  ray  will  be 
reflected  back  into  the  denser  medium.  This  angle  of  in- 
cidence is  called  the  critical  angle , and  the  reflection  begin- 
ning at  this  angle  is  called  total  reflection.  The  critical  angle 

0 * 

evidently  depends  upon  the  values  of  the  index  of  refraction 

for  the  media,  and  as  these  vary  with  the  wave  length  of 

the  incident  light,  the  critical  angle  also  varies  with  the 

% 

color  of  the  light.  We  may  however  determine  an  average 
value  for  white  light. 

-When  the  angle  of  incidence  has  reached  its  critical  value 
a,  the  angle  of  refraction  is  90°,  and  by  Snell's  law  we  have 
for  the  index  of  refraction  {n  being  always  taken  for  incidence 
in  the  rarer  medium,) 

l/n  = sin  a/  sin  90° , or  sin  a — 1/n. 

Hence  if  we  measure  the  index  of  refraction  of  a substance 
we  may  calculate  its  critical  angle  by  means  of  this  relation 
or  vice  versa. 

Directions. — Place  the  glass  cube  ABCD  near  the  right 
hand  upper  edge  of  a sheet  of  paper  on  the  table,  and  draw 
its  outline  with  a sharp  pencil.  (See  Fig.  68.)  Stick  a ver- 
tical pin  P in  the  table  close  to  the  glass  and  near  its  corner 
B.  The  glass  must  not  be  moved  from  this  position  during 
the  experiment.  Lay  the  straight  edge  on  the  paper  in 
front  of  the  glass  and  place  it  so  that  when  sighting  along 
its  edge  the  pin  can  be  seen  through  the  glass.  Then  draw 
a straight  line  FE  along  its  edge.  Shift  the  ruler  to  a dif- 
ferent position  at  which  the  pin  may  be  seen  along  its  edge, 
and  in  a similar  manner  draw  five  such  lines  as  FE.  Remove 
the  glass  and  prolong  these  lines  to  meet  DC.  Then  join 
the  point  P to  the  intersection  of  each  lin^  with  DC,  and 
prolong  these  incident  rays  to  some  convenient  length,  say 


In  General  Physics 


267 


10  cm.  The  index  of  refraction  may  then  be  determined 
for  each  pair  of  incident  and  refracted  rays  as  follows: 

Lay  off  an  equal  length  (10  cm.)  along  the  refracted  ray 
EF , draw  the  normal  to  the  surface  DC  at  the  intersection 
E , and  drop  perpendiculars  from  the  extremities  F and  G 


of  these  lengths  upon  the  normal  NE.  The  sine  of  the  angle 
of  incidence  is  then  GH/GE , (for  the  angle  GEH  equals  the 
angle  PEN.)  and  the  sine  of  the  angle  of  refraction  is 
FI/FE  or  FI/GE.  Hence  the  index  of  refraction  is  by 
SnelPs  law 

n — sin  r/sin  i = FI/GH. 

From  the  mean  value  of  n calculate  the  value  of  a by 
the  equation  given. 


268 


Manual  of  Experiments 


Questions. — a.)  Explain  why  a right  angled  prism  may 
be  used  to  change  the  direction  of  a beam  of  light  at  right 
angles  to  its  original  direction,  when  it  impinges  perpen- 
dicularly upon  one  of  the  smaller  faces,  b.)  Would  the 
prism  act  in  the  same  way  if  it  were  immersed  in  carbon 
bisulphide  (n  = 1.63)? 

ARRANGEMENT  OF  LENSES  IN  MICROSCOPES 

AND  TELESCOPES. 

Theory  and  Definitions. — (D.  700-705;  G.  597-608; 
K.  882-891;  W.  353-355.)  Any  convex  (convergent)  lens 

9 

may  be  used  as  a simple  microscope  or  magnifying  lens. 
With  the  unaided  eye  an  object  is  most  distinctly  seen  at  a 
distance  of  about  25  cm.  When  the  object  is  observed 
through  the  lens  placed  at  less  than  its  focal  length  from  the 
object,  the  angle  which  it  subtends  at  the  eye  is  increased 
and  a magnified  virtual  image  is  seen.  The  magnifying 
power  is  defined  as  the  ratio  of  the  apparent  size  of  the  image 
to  the  size  of  the  object  as  seen  at  the  distance  of  most  dis- 
tinct vision,  (25  cm.).  Though  the  object  is  always  a little 
nearer  the  lens  than  the  focus,  if  the  eye  is  placed  very  near 
the  lens,  the  approximate  value  of  the  magnifying  power 
will  be  25  cm.  divided  by  the  focal  length  /. 

In  the  compound  microscope  at  least  two  convex  lenses 
are  used.  One  of  these,  the  objective,  is  placed  so  that  the 
object  lies  just  beyond  its  principal  focus  and  it  forms  an 
inverted  and  magnified  real  image.  The  second  lens,  the 
eye  piece,  is  placed  so  that  this  real  image  falls  just  inside 
of  its  focus,  and  forms  a more  magnified  virtual  image  of 
the  first  inverted  image.  The  magnification  by  the  objective 
is  the  ratio  of  image  to  object  distance  v/u,  and  that  by  the 
eye  piece  is  2 5/f,  whence  the  total  magnification  is  25 v/fu. 


In  General  Physics 


269 


The  astronomical  telescope  consists  of  at  least  two  convex 
lenses.  The  objective  forms  an  inverted  reduced  real  image 
of  a distant  object.  This  image  when  viewed  through  the 
eye  piece  appears  as  a magnified  virtual  image  which  is 
still  inverted.  To  form  an  erect  image  one  or  more  additional 
lenses  would  be  needed. 

In  the  Galilean  telescope  the  image  formed  by  the  convex 
objective  is  viewed  through  a concave  (divergent)  eyepiece, 
which  is  so  placed  in  front  of  the  image  that  the  distance  be- 
tween the  two  lenses  is  the  difference  between  their  focal 
lengths. 

The  magnification  in  either  kind  of  telescope  is  the  ratio 
F/f  where  F is  the  focal  length  of  the  objective,  f that  of 
the  eyepiece. 

In  practical  microscopes  and  telescopes  both  eyepiece 
and  objective  are  built  up  of  several  lenses  in  order  to  give 
the  proper  focal  length  and  to  overcome  spherical  and  chro- 
matic aberration,  etc.,  the  lenses  being  mounted  in  tele- 
scoping tubes. 


50.  MICROSCOPE  AND  TELESCOPE. 

a.  To  find  the  magnifying  power  of  a simple  microscope. 

b.  To  arrange  a pair  of  lenses  for  use  as  a compound 
microscope. 

c.  To  arrange  a pair  of  lenses  for  use  as  an  astronomical 
telescope. 

d.  To  arrange  three  lenses  for  use  as  a terrestrial  tele- 
scope. 

e.  To  arrange  a pair  of  lenses  for  use  as  a Galilean  tele- 
scope. 


270 


Manual  of  Experiments 


Apparatus. — Optical  bench  sliding  screen  with  fine 
printing  on  it,  screened  light,  one  large  double  convex  lens, 
two  small  convex  lenses,  one  small  double  concave  lens, 
holder  for  each  lens,  sliding  cross-hairs,  millimeter  scale,  pins. 

Directions. — a. — Place  the  large  convex  lens  at  one  end 
of  the  bench.  Adjust  the  mounted  cross  hairs  behind  the 
lens  until  they  are  sharply  focussed.  Place  the  screen  at 
a distance  of  25  cm.  behind  the  lens.  Looking  at  the  parallel 
sides  of  the  opening  in  the  cardboard  on  which  cross  hairs 
are  mounted,  through  the  lens  anduat  the  screen  above  the 
lens  place  pins  in  the  screen  at  the  position  of  these  parallel 
sides.  Measure  the  distance  between  lens  and  cross  hairs 
and  between  lens  and  screen,  also  width  of  opening  in  cross 
hair  mount  and  distance  between  pins.  The  width  of  the 
image  may  be  measured  more  directly  by  placing  a mm. 
scale  in  position  of  the  screen  and  observing  the  cross  hairs 
and  slit  through  the  lens  with  one  eye,  the  scale  through 
the  air  with  the  other  eye.  The  position  of  the  scale  should 
be  adjusted  until  the  image  is  seen  apparently  projected  on 
the  scale,  so  that  the  position  of  the  sides  of  the  slit  on  the 
scale  can  be  noted. 

The  ratio  of  the  image  width  to  the  object  width  is  the 
magnification  at  the  distance  of  distinct  vision.  Compare 
this  with  the  ratio  between  the  distance  from  lens  to  screen 
and  from  lens  to  cross  hairs.  Repeat  the  above  measure- 
ments. 

b.  Compound  Microscope. — Determine  the  focal  lengths 
of  the  different  lenses.  (See  Exp.  47).  Place  a small  convex 
lens  <2,  at  a little  more  than  its  focal  length  from  the  slit  in 
the  light  screen,  using  it  as  the  objective.  Adjust  the  sliding 
screen  until  an  image  of  the  slit  is  sharply  outlined  on  it. 
Measure  the  length  of  both  the  slit  and  its  image,  and  the 


In  General  Physics  271 

object  and  image  distances  from  the  lens  (u  and  v).  Note 

whether  image  is  erect  or  inverted.  Place  the  other  small 

/ 

convex  lens  b , (the  eyepiece),  back  of  the  screen  at  a distance 
less  than  its  focal  length.  Remove  the  screen,  adjust  the 
lens  b until  the  image  seen  through  it  is  sharply  outlined 
and  note  whether  it  is  erect  or  inverted.  Clamp  the  meter 
stick  in  a vertical  position  25  cm.  back  of  lens  b and  just  be- 
side the  bench  so  that  it  does  not  interfere  with  the  light 
between  the  lenses.  With  one  eye  look  through  the  lens  at 
the  slit,  then  open  the  other  and  look  through  air  at  the  meter 
stick,  adjusting  the  position  of  the  latter  until  the  image 
of  the  slit  is  seen  apparently  projected  on  the  scale.  The 
ratio  of  the  image  length  at  25  cm.  to  the  slit  length  is  the 
magnification.  Compare  this  with  the  magnification  calcu- 
lated from  u,  v and  f (focal  length  of  eye  piece).  Compare 
the  length  of  slit  with  that  of  its  two  images.  Observe 
that  the  image  formed  by  b cannot  be  projected  upon  a screen. 
Repeat  the  above  measurements  using  the  large  convex 
lens  c for  the  objective. 

c.  Astronomical  Telescope. — Place  the  large  convex  lens 
c about  60  to  100  cm.  from  the  slit  and  locate  the  position 
of  the  image  by  means  of  the  screen.  Measure  the  lengths 
of  slit  and  image  and  note  whether  the  image  is  erect  or  not. 
Place  the  convex  lens  a back  of  the  image  at  less  than  its 
focal  length,  remove  the  screen  and  observe  the  image 
through  the  lens  a.  Note  whether  the  image  formed  by  a 
is  erect  of  not.  Adjust  a until  the  image  appears  sharp, 
and  measure  the  length  from  c to  a.  Place  the  cross  hairs 
back  of  a and  adjust  them  until  their  image  appears  sharp. 
Compare  the  position  of  the  cross  hairs  with  the  position 
of  the  image  formed  by  c. 

To  determine  the  magnification  place  a vertical  meter 
stick  from  5 to  10  meters  in  front  of  lens  c and  observe  it 


272 


Manual  of  Experiments 


and  its  image  simultaneously  as  explained  for  the  microscope. 
Adjust  the  meter  stick  until  its  image  is  projected  upon  it, 
and  note  the  position  on  the  scale,  of  two  marks  on  the  image. 
The  ratio  of  the  distance  between  the  two  positions  to  that 
between  the  two  marks  will  be  the  magnification.  Compare 
this  with  the  magnification  calculated  from  the  focal  lenghts 
of  objective  and  eyepiece. 

d . Terrestrial  telescope. — Leaving  c in  position  move  a 
about  twice  its  focal  length  from  the  image  formed  by  c. 
Observe  that  a real  image  is  now  formed  by  a . Note  its 
position  and  length  and  whether  it  is  erect  or  not.  Place 
the  third  convex  lens  b back  of  the  second  image  at  a distance 
less  than  its  focal  length.  Observe  the  image  through  b 
adjusting  the  latter  until  it  forms  a sharp  image  and  note 
whether  this  image  is  erect  or  not.  Measure  the  length  from 
dob.  Determine  the  magnification  as  before,  and  compare 
the  lengths  of  the  slit  and  the  three  images. 

e.  Galilean  telescope. — Leave  c in  position,  but  remove 
a and  b.  Place  the  concave  lens  d in  front  of  the  image 
formed  by  c.  Observe  that  no  real  image  is  formed  by  d. 
Adjust  d until  the  image  seen  through  it  is  sharp.  Measure 
the  length  from  c to  d,  and  compare  it  with  the  lengths  of 
the  other  two  telescopes.  Determine  the  magnifying  power 
as  with  the  astronomical  telescope  and  compare  it  with  the 
value  calculated  from  the  focal  lengths  of  the  lenses. 

Problem. — Draw  diagram^showing  the  formation  of 
the  images  in  each  of  the  above  five  cases. 


In  General  Physics 


273 


51.  THE  SPECTROMETER. 

INDEX  OF  REFRACTION  OF  A PRISM. 

To  determine  the  index  of  refraction  of  a glass  prism 
from  the  prism  angle  and  angle  of  minimum  deviation,  by 
means  of  a spectrometer.  (D.  659,  660,  709;  K.  557-561; 
G.  835,  847,  848;  W.  345-347,  357,  367,  369.) 

Apparatus. — Spectrometer  and  prism,  incandescent 
lamp,  hood,  and  black  cloth. 

Theory;  Definitions  and  Description.— The  index  of 
refraction  at  the  interface  between  two  media  is  the  ratio  of 
the  velocities  of  light  in  the  two  media,  or  the  ratio  of  the 
sine  of  the  angle  of  incidence  to  the  sine  of  the  angle  of  re- 
fraction. This  ratio  is  according  to  Snell's  law  a constant 
for  all  ordinary  substances  for  a given  wave  length  of  light. 
As  usually  tabulated  this  index  is  given  for  different  sub- 
stances in  air,  the  angle  of  incidence  being  taken  in  air.  If 
a prism  of  the  substance  is  available,  its  index  of  refraction 
may  be  determined  with  the  aid  of  a spectrometer. 

Let  A be  the  angle  of  the  prism  between  the  two  faces 
which  intersect  in  the  refracting  edge  and  D the  angle  of 
minimum  deviation,  i.  e .,  the  least  angle  which  is  formed  by 
any  ray  between  its  direction  before  entering  the  prism  and 
its  direction  after  leaving  the  prism.  It  may  be  shown  that 
the  index  of  refraction  is 

A+D  / A 

n = sin — sin — 

2 / - 2 

A and  D may  be  measured  on  the  spectrometer.  The  spec- 
trometer consists  of  an  achromatic  telescope,  a graduated 
circle,  and  a collimator  supported  on  a heavy  tripod  base. 
The  collimator  is  a device  for  producing  a beam  of  parallel 


274  Manual  of  Experiments 

rays.  It  consists  of  an  adjustable  slit  and  a convex  lens, 
at  opposite  ends  of  two  telescoping  tubes.  In  some  forms 
of  spectrometer  the  three  essential  parts  of  the  instrument 
may  be  rotated  about  a common  center  and  may  be  clamped 
in  any  position  relative  to  each  other.  Tire  collimator  and 
telescope  may  also  be  rotated  about  vertical  and  horizontal 
axes  perpendicular  to  their  optical  axes  and  are  provided 
with  slow  motion  tangential  screws  for  adjusting  their  po- 
sition accurately.  The  circular  scale  is  provided  with  two 
verniers  and  reading  glasses  180°  apart,  for  accurate  reading. 
A small  adjustable  table  at  the  center  of  the  graduated  circle 
serves  to  hold  the  prism,  grating  or  other  device  to  be  used. 

Directions. — To  Adjust  the  Spectrometer. — Focus  the 
telescope  for  parallel  light  by  placing  the  cross  hairs  in  an  ob- 
lique position,  adjusting  the  eyepiece  until  they  appear 
sharp,  and  then  sighting  the  telescope  on  a distant  object, 
adjusting  the  distance  between  eyepiece  and  objective 
(without  altering  the  relative  position  of  eyepiece  and  cross 
hairs)  until  there  is  no  parallax  between  the  intersection  of 
the  cross  hairs  and  the  image  of  the  distant  object. 

Next  level  the  instrument,  remove  the  prism  table,  and 
place  a vertical  lead  pencil  in  its  stead.  Adjust  the  telescope 
and  collimator  until  their  axes  are  in  the  same  horizontal 
plane  and  in  the  vertical  plane  through  the  center  of  pencil. 
When  this  is  properly  done  the  intersection  of  the  cross  hairs 
will  lie  on  the  center  of  the  slit.  If  it  is  necessary  to  rotate 
either  instrument  in  furtheu-adjustments,  be  careful  to 
grasp  them  by  their  arms  not  by  their  tubes,  so  that  they  are** 
not  accidentally  turned  about  their  own  vertical  axis. 
Next  place  a light  (incandescent  lamp)  behind  the  slit  and 
sight  through  telescope  and  collimator  at  it.  Let -the  slit 
be  quite  narrow  but  distinctly  visible  and  adjust  the  distance 


In  General  Physics 


275 


between  it  and  the  collimator  lens  until  there  is  no  parallax 
between  slit  and  cross  hairs  of  the  telescope.  The  prism 
table  is  then  replaced  and  the  prism  set  in  position  with 
its  refracting  edge  parallel  to  the  axis  of  the  spectrometer 
and  its  center  in  line  with  the  axis  of  telescope.  The  planes 
of  the  vertical  faces  of  prism  must  be  as  nearly  perpen- 
dicular as  possible  to  the  plane  passing  through  the  op- 
tical axes  of  the  lens  systems  of  telescope  and  collimator. 
Ask  the  instructor  for  assistance  in  doing  this. 

To  measure  the  angle  of  the  prism. — To  avoid  possible 
errors  in  interpreting  readings,  clamp  the  zero  of  the  circular 
scale  under  the  collimator  C.  (See  Fig.  69.)  Turn  the 
prism  P until  its  refracting  edge  is  pointed  directly  at  the 
center  of  the  collimator  lens  and  clamp  it  in  position  with 
this  edge  over  the  center  of  the  spectrometer.  Illuminate 
the  slit  by  means  of  the  incandescent  lamp  and  adjust  its 
width  so  that  it  is  very  narrow,  yet  distinct.  Turn  the 
telescope  T about  the  center  of  the  circular  scale,  until  the 
image  of  the  slit  reflected  from  the  right  face  of  the  prism 
is  seen.  Clamp  the  telescope  in  position  and  by  means  of 
the  tangential  screw  set  the  cross  hairs  on  the  right  edge  of 
this  image.  Read  the  position  on  the  verniers.  Repeat  this 


276 


Manual  of  Experiments 


performance  with  the  image  reflected  from  the  left  face  of 
the  prism,  setting  the  cross  hairs  on  the  right  edge  as  before. 
It  may  be  shown  that  the  angle  A is  half  the  angle  between 
the  two  positions  of  the  telescope.  Alter  the  position  of  the 
prism  slightly  and  repeat  this  determination  as  a check. 

To  measure  the  angle  of  minimum  deviation. — Place  the 
prism  with  its  center  over  the  center  of  the  spectrometer, 
(the  base  being  nearer  to  the  center  than  the  refracting 
edge,)  and  clamp  it  in  position.  If  the  beam  of  light  is 
incident  at  the  angle  of  minimum  deviation  its  path  through 
the  prism  will  be  symmetrical  with  respect  to  the  two  polished 
faces  of  the  prism.  When  the  slit  is  illuminated  by  the  in- 
candescent lamp  the  prism  forms  a spectrum.  To  get  a 
sharply  defined  image  of  the  slit  it  is  necessary  to  use  a source 
of  monochromatic  light  (of  a single  wave  length)  such  as 
the  sodium  flame.  Remove  the  lamp  and  illuminate  the 
slit  with  a Bunsen  burner,  on  the  tip  of  which  has  been  placed 
an  asbestos  tube  impregnated  with  sodium  chloride.  An 
intense  yellow  flame  is  produced.  Turn  the  prism  table 
until  the  refracting  edge  points  at  right  angles  to  the  colli- 
mator tube  and  locate  the  refracted  image  of  the  slit  by  eye. 
Turn  the  telescope  until  the  image  may  be  observed  through 
it.  Then  gradually  turn  the  refracting  edge  aw^ay  from  the 
collimator.  The  image  will  turn  toward  the  collimator. 
Follow  the  image  with  the  telescope  until  it  just  begins  to 
move  in  the  opposite  direction.  Clamp  the  telescope  in 
this  position,  and  turn  the  prism  table  slightly  first  in  one 
direction,  then  in  the  otherT^until  the  image  is  just  at  its 
turning  point.  Then  adjust  the  cross  hairs  on  the  image 
by  means  of  the  tangential  screw,  and  read  the  verniers. 
Next  remove  the  prism,  place  the  telescope  in  line,  with  the 
collimator,  clamp  it,  adjust  the  cross  hairs  to  coincide  with 
the  image  of  the  slit  by  means  of  the  tangential  screw  and 


In  General  Physics 


277 


read  the  verniers.  The  difference  between  the  two  positions 
of  the  telescope  is  the  angle  of  minimum  deviation  D.  Alter 
the  position  of  the  circular  scale  slightly  and  repeat  the  de- 
termination. 

v Calculate  the.  value  of  n from  the  mean  values  of  A 
and  D. 

Problems. — 1.— Draw  a diagram  and  show  the  relation 
between  A and  the  telescope  positions  for  determining  it. 

2. — Draw  a diagram  and  derive  the  expression  for  the 
index  of  refraction  as  given  under  theory. 

52.  THE  DIFFRACTION  GRATING. 

i< 

To  determine  the  wave  length  of  the  D lines  of  the  sodium 
spectrum  with  the  aid  of  a diffraction  grating.  (D.  696; 
G.  660-663;  K.  935-940;  W.  267,  374.) 

Apparatus. — Spectrometer,  diffraction  grating,  sodium 
light,  incandescent  light  and  black  cloth. 

Theory  and  Method.— The  diffraction  grating  is  a plate 
of  glass  or  of  polished  speculum  metal  on  which  a series  of 
fine  parallel  lines  (usually  several  thousand  in  the  space  of 
a centimeter)  are  ruled.  When  light  passes  through  the 
glass^or  is  reflectedjjpm  the  metal  these  lines  form  spectra 
by  diffraction.  A replica  of  these  gratings  in  gelatine  or 
other  transparent  substance,  placed  between  two  parallel 
sheets  of  plate  glass,  acts  in  the  same  way.  The  method  of 
producing  the  spectra  may  be  explained  with  the  aid  of 
figure  70,  in  which  the  cross  hatched  spaces  represent  the 
opaque  ruled  lines  and  the  spaces  between  them  are  trans- 
parent glass  or  gelatine.  If  a plane  wave  of  monochromatic 
light  from  a narrow  slit,  placed  parallel  to  the  ruled  lines, 


278 


Manual  of  Experiments 


falls  on  the  grating,  the  spaces  A and  B become  sources  of 
new  wavelets  which  travel  out  in  all  directions  behind  the 
grating  and  may,  by  means  of  a lens,  be  brought  to  a focus 
on  a screen  SS.  The  point  P on  the  screen  is  equidistant 
from  A and  i?,  and  wavelets  traveling  along  the  paths  AP 
and  BP  are  in  the  same  phase  and  form  a new  wave  front 
at  equal  distances  from  A and  B which,  when  brought  to 
a focus  at  P forms  an  image  of  the  slit.  This  is  called  the 
central  image.  For  some  point  R on  the  screen,  the  distance 
AR  is  greater  than  the  distance  BR  by  an  amount  AD  which 
is  equal  to  one  wave  length,  sy>  that  the  wavelets  at  D and 
B being  in  the  same  phase,  form  a new  wave  front.  When 
this  is  brought  to  a focus  at  R it  produces  an  image  of  the 
slit  called  the  first  order  spectrum.  Similarly  for  some  point 
on  the  screen  beyond  R a new  wave  front  is  formed  by  a 
wavelet  just  starting  from  B and  one  from  A which  has 
passed  over  two  wave  lengths.  This  produces  on  the  screen 
the  second  order  spectrum;  and  so  on  for  higher  orders. 
An  exactly  similar  set  of  images  is  formed  in  the  same  way 
on  the  other  side  of  P.  For  all  points  between  these  images 
there  is  interference  between  the  wavelets  from  A and  i?, 
and  more  especially  when  the  path  difference  from  A and  B 
is  an  odd  multiple  of  a half  wave  length,  the  waves  are  in 
opposite  phase,  and  complete  interference  results  in  a dark 
“image”  or  space  on  the  screen.  The  same  explanation 
applies  in  the  case  of  more  than  two  ruled  lines,  but  the 
spectra  in  this  case  are  much  brighter  because  of  the  greater 
number  of  lines  from  which-The  light  comes.  If  in  place  of 
monochromatic  light  we  use  white  light  the  central  image 
will  be  white,  because  all  wave  lengths  travel  over  equal  dis- 
tances to  this  point.  But  the  distance  AD  will  differ  with 
the  wave  lengths,  consequently  the  first  order  images  will 
not  all  be  formed' at  i?,  but  at  some  point  nearby.  As  a 


In  General  Physics 


279 


result  a continuous  spectrum  of  all  the  colors  will  be  formed 
in  the  neighborhood  of  R , with  the  violet  (due  to  the  shortest 
wave  lengths)  nearest  to  the  central  image.  The  same  is 
true  of  the  spectra  of  higher  orders. 

Let  the  distance  AB  between  adjacent  rulings  be  <2,  the 
difference  of  path  AD  be  X,  and  the  angle  ABD  be  0.  Then 
from  triangle  ABD  we  have 

\ = a sin  9 (1) 


or  if  the  angle  of  the  nth  order  spectrum  be  <j> 

n\=  a sin  <£. 


(2) 


As  the  triangles  ABD  etc.,  are  not  exactly  right  triangles 
these  equations  are  not  quite  true,  but  for  the  lower  orders 


280 


Manual  of  Experiments 


of  spectra  they  may  be  used  without  making  an  appreciable 
error. 

If  the  grating  is  placed  on  a spectrometer  table,  the 
angle  d may  be  measured  directly.  The  number  of  lines 
per  centimeter  is  usually  marked  on  the  grating  so  that  a 
is  easily  calculated.  From  these  we  calculate  the  value  of  X. 

If  a spectrometer  is  not  available,  we  may  fix  a scale  in 
front  of  the  grating,  and  move  a telescope  along  it  to  measure 
the  distance  PR  between  images.  The  distance  RC  (or  PC, 
between  grating  and  scale,  since  these  are  pactically  equal) 
is  readily  measured  and  the  sine  of  6 will  be  PR/RC . 

Directions. — Focus  the  eyepiece  of  the  telescope 
on  the  cross  hairs  and,  having  focussed  the  telescope 
for  parallel  light,  focus  the  slit  of  the  collimator  so  that  there 
is  no  parallax  between  the  image  of  the  slit  and  the  cross 
hairs  of  the  telescope.  (See  Exp.  51).  Place  the  grating 
in  the  holder  on  the  platform  so  that  the  ruled  side  of  the 
grating  is  turned  away  from  the  collimator.  See  that  the 
grating  lines  are  vertical  and  the  plane  of  the  grating  is  at 
right  angles  to  the  axis  of  the  collimator.  Make  the  slit  of 
the  collimator  very  narrow  and  illuminate  it  with  a sodium 
flame.  Place  the  cloth  over  the  spectrometer  to  cut  off 
external  light.  Move  the  telescope  so  as  to  be  in  line  with 
the  collimator  and,  having  the  cross  hairs  on  the  image  of 
the  slit,  observe  the  readings  of  the  vernier.  All  parts  of 
the  spectrometer,  except  the  telescope,  should  be  claipped. 

Move  the  telescope  a few  degrees  to  one  side  and  adjust 
it  till  the  cross  hairs  are  on  the  image  of  the  slit  as  seen  in 
the  first  order  spectrum.  The  slit  must  be  vertical  and  parallel 
to  the  rulings  on  the  grating  for  best  definition.  If  the  line 
is  not  readily  found,  an  incandescent  light  may  be  placed 
in  front  of  the  slit  which  will  produce  a continuous  spectrum 


In  General  Physics 


281 


that  is  readily  located  and  then  the  sodium  light  substituted 
for  it.  Note  the  readings  of  the  verniers  in  degrees,  minutes, 
and  seconds  of  arc,  when  the  cross  hairs  are  on  the  first 
order  spectrum,  and  find  the  mean  of  the  differences  be- 
tween corresponding  vernier  readings  for  the  first  order 
spectrum  and  the  central  image.  This  is  one  value  of  6. 
Move  the  telescope  to  the  other  side  of  the  central  image 
and  make  a second  determination  of  the  angle  6 by  adjust- 
ing the  telescope  so  that  the  image  of  the  first  order  spectrum 
on  that  side  is  in  coincidence  with  the  cross  hairs.  Repeat 
each  determination  two  times  and  find  the  mean  of  the  six 
values  of  6 (three  on  one  side  and  three  on  the  other  side,) 
the  angle  through  which  the  telescope  must  be  turned  from 
its  position  of  coincidence  with  the  central  image  to  its  po- 
sition of  coincidence  with  the  first  order  spectra.  Note  the 
number  of  lines  to  the  centimeter  as  recorded  on  the  grating 
and  determine  the  distance  a between  two  adjacent  lines. 
Substitute  the  values  of  a and  6 in  equation  (1)  and  calcu- 
late X the  wave  length  of  sodium  light. 

In  a similar  manner  make  three  determinations  of  the 
angle  </>  for  coincidence  of  the  cross  hairs  with  the  image  of 
the  second  order  spectra  on  each  side  of  the  central  image, 
and  find  the  mean  of  the  six  values  of  4>  thus  observed.  The 
second  order  spectrum  will  be  at  an  angle  about  twice  the 
angle  for  the  first  order.  It  will  be  rather  indistinct,  and 
consequently  care  must  be  taken  to  have  the  slit  well  illu- 
minated and  all  external  light  cut  off  the  grating  as  much 
as  possible.  Substitute  the  value  of  <£  found  for  the  second 
order  spectra  together  with  the  value  of  a in  equation  (2) 
where  n has  the  value  2,  and  calculate  the  wave  length  X 
of  sodium  light. 

By  careful  adjustments  it  will  be  possible  to  see  that  the 
sodium  line  is  composed  of  two  lines,  and  especially  so  in 


282 


Manual  of  Experiments 


the  second  order  spectra.  Replace  the  sodium  light  by  the 
incandescent  lamp  and  observe  the  order  of  the  colors  in  the 
spectra  of  both  the  first  and  second  orders  on  each  side  of 
the  central  image.  Observe  which  spectra  are  the  most 
distinct  and  which  spread  out  the  most.  Make  a sketch 
in  your  report  showing  the  order  of  the  colors  in  diffraction 
grating  spectra. 

Questions. — 1. — How  does  the  order  of*  the  colors  in  a 
grating  spectrum  differ  from  the  order  in  a prism  spectrum? 

2. — What  is  the  frequency  of  the  vibration  which  pro- 
duces a yellow  sodium  line? 


EXPERIMENTS  IN  SOUND. 

Theory  and  Definitions. — Sound  is  transmitted  by 
means  of  longitudinal  waves,  the  direction  of  vibration  of 
the  particles  of  the  transmitting  medium  being  parallel  to 
the  direction  of  propagation  of  the  wave.  The  wave  is 
transmitted  as  a series  of  alternate  condensations  and  rare- 
factions of  the  medium.  The  length  of  the  wave  is  the  dis- 
tance from  any  one  vibrating  particle  to  the  next  succeeding 
(or  preceding)  one  which  has  the  same  phase.  It  is  there- 
fore the  distance  between  two  successive  points  of  maximum 
condensation  or  of  maximum  rarefaction. 

The  velocity  of  the  wave  transmission  is  the  wave  length 
times  the  frequency  of  the  vibration  producing  it.  The 
pitch  of  the  sound  depends  upon  the  frequency,  the  intensity 
upon  the  amplitude,  and  the  timbre  upon  the  shape  of  the 
wave. 

When  waves  impinge  upon  an  elastic  body  which  has  the 
same  natural  period  or  the  same  frequency  as  the  wave,  the 


In  General  Physics 


283 


body  is  set  in  sympathetic  vibrations  and  there  is  said  to  be 
resonance  between  the  two. 

When  two  bodies  vibrating  near  each  other  have  very 
nearly  the  same  natural  period,  pulsations  of  sound  or  beats, 
are  produced  due  to  the  alternate  reinforcement  and  inter- 
ference of  the  waves.  Two  bodies  may  be  brought  to  the 
same  pitch  by  altering  the  period  of  one  of  them  until  the 
beats  disappear  and  a sound  of  uniform  intensity  is  heard. 

When  a column  of  air  in  a tube  closed  at  one  end,  is  set 
in  vibration  by  resonance,  there  is  a condensation  at  the 
closed  end  or  a node  is  formed  there  and  a rarefaction  at  the 
open  end  producing  an  antinode.  The  shortest  closed  tube 
in  which  resonance  could  be  produced  by  a given  wave  would 
have  a length  equal  to  one-quarter  of  the  wave  length.  The 
next  length  in  which  resonance  would  occur  would  be  three- 
quarters  of  the  wave  length,  etc.  In  an  open  tube  resonance 
would  occur  for  lengths  equal  to  multiples  of  a half  wave 
length. 

When  a stretched  string  is  set  in  vibration,  the  wave 
length  of  the  sound  produced  cannot  be  greater  than  twice 
the  length  of  the  string  since  there  must  be  a node  at  each 
end. 

It  may  be  shown  that  the  frequency  of  vibration  or  the 
pitch  of  the  sound  produced  is  inversely  proportional  to  the 
length  of  string  and  to  the  square  root  of  its  mass  per  unit 
length,  but  directly  proportional  to  the  square  root  of  its 
tension.  If  T represents  tension,  /,  length  of  string,  and  m, 
mass  per  unit  length,  the  frequency  is 

1 /T 

n = — / — 

2/  V m 


284 


Manual  of  Experiments 


53.  GRAPHICAL  DETERMINATION  OF  PITCH  OF 

TUNING  FORKS. 

To  determine  the  pitch  of  two  tuning  forks  graphically. 
(D.  592,  609;  G.  245;  K.  304;  W.  267,  288.) 

Apparatus. — Two  tuning  forks  to  be  tested,  a long 
board  slide-way  with  slides  and  clamps  for  pendulum  and 
tuning  fork,  strip  of  plate  glass,  rubber  hammer  for  fork, 
steel  dividers,  paste  of  whiting  or  chalk  dust  and  dauber. 

Description  and  Method. — The  slide-way  is  a smooth 
board  about  5 feet  long  and  6 inches  wide,  provided  with 
side  guard  strips.  The  wooden  slide — about  2 feet  long — 
slips  easily  in  the  slide-way.  A tuning  fork  and  a small 
pendulum  are  supported  above  the  slide-way  so  that  they 
may  vibrate  across  the  path  of  the  slider.  The  clamps  allow 
an  adjustment  of  the  pendulum  and  the  fork.  The  strip 
of  glass  may  be  attached  to  the  top  of  the  slide  by  appropriate 
fasteners. 

The  method  used  is  to  get  traces  of  the  vibrating  fork 
and  swinging  pendulum  in  the  same  line  on  the  glass  and  to 
count  the  number  of  waves  of  the  fork  in  one  vibration  or 
wave  length  of  the  pendulum;  then  by  timing  the  pendulum 
vibration,  to  determine  the  number  of  vibrations  of  the  fork 
in  one  second,  i.  <?.,  its  pitch,  by  dividing  the  ^number  of 
oscillations  of  the  fork  corresponding  to  one  oscillation  of 
the  pendulum  by  the  time  required  for  the  latter. 

Directions. — Take  the  wooden  slide  and  attached  strip 
of  glass  to  the  sink  and  clean  the  glass  thoroughly  with  a 
dry  cloth.  Cover  the  glass  with  a uniform  coating  of  whiting 
paste  and  let  it  dry.  Clamp  the  pendulum  and  the  tuning 
fork  in  place,  so  that  the  vibrating  end  of  the  fork  is  near 
the  pendulum.  Fasten  a stylus  of  copper  foil  at  the  end  of 


In  General  Physics 


285 


each,  so  that  if  the  slide  were  pushed  through,  the  lines  traced 
would  coincide  approximately.  It  is  best  to  have  the  styluses 
touch  the  glass  lightly  and  slant  in  the  same  direction  that 
the  slide  is  to  pass.  Move  the  slide  back  until  the  styluses 
are  free  to  vibrate,  then  set  the  pendulum  vibrating  through 
a small  arc,  and  strike  the  fork  a single  sharp  blow.  Pass 
the  slides  through  at  a moderate  rate  so  that  the  pendulum 
traces  one  or  more  complete  oscillations.  Count  the  number 
of  oscillations  of  the  fork  in  a complete  oscillation  of  the  pen- 
dulum. In  order  to  compare  the  fork  trace  with  the  pen- 
dulum trace  made  at  the  same  time,  it  is  necessary  to  shift 
the  crossing  point  of  the  pendulum,  trace  the  distance  be- 
tween the  two  styluses.  This  is  because  one  stylus  makes 
its  trace  on  the  glass  before  the  other  touches  the  glass. 
See  Fig.  71  where  the  dotted  line  shows  the  shifted  position 
for  the  pendulum  trace.  Repeat  the  experiment  until  two 
values  of  the  fork  oscillations  are  found  that  differ  by  two 
or  less.  Repeat  for  the  second  fork.  Observe  the  time  for 
100  complete  oscillations  of  the  pendulum,  and  calculate  the 
time  for  one  oscillation.  Repeat  as  a check.  Calculate  the 
number  of  oscillations  for  each  fork  in  one  second. 

Question  — Describe  exactly  how  your  results  are 
affected  by  the  friction  of  the  styluses  on  the  glass. 


Fig.  71 


286 


Manual  of  Experiments 


54.  VIBRATIONS  OF  STRETCHED  STRINGS. 

To  test  the  relation  between  frequency  of  vibration  of  a 
stretched  string  and  its  length,  tension  and  mass.  (D.  602; 
G.  266-270;  K.  326,  327;  W.  267,  276.) 

Apparatus. — Sonometer  box,  two  tuning  forks,  two  steel 
wires  of  different  cross  sections,  two  masses  for  stretching 
wires. 

Description. — The  sonometer  box  is  a light  pine  box 
about  4 feet  long,  3 inches  high,  and  4 inches  broad.  See  Fig. 
72.  It  is  used  to  strengthen  the  feeble  sound  of  the  wire.  The 
box  rests  on  rubber  feet  to  prevent  the  communication  of  the 
vibrations  to  the  table.  On  top  of  the  box  is  a fixed  fret  and 
a moveable  one,  also  a quarter  circle  of  brass  that  rocks  in 


a V-groove  of  the  same  metal  at  the  end  of  the  box.  The 
wire  is  fastened  to  a brass  post  at  one  end  of  the  box,  and 
passes  over  both  frets  and  the  small  groove  of  the  quarter 
circle.  The  stretching  mass  is  hung  from  a ring  at  the 
end  of  the  wire. 


In  General  Physics 


287 


Directions. — Relation  between  pitch  of  wire  and  length. — 
Use  the  smaller  wire  and  hang  the  larger  iron  block  (about 
4,000  gms.)  to  the  free  end  of  the  wire.  See  that  the  quarter 
circle  rocks  freely  in  the  groove  so  that  the  suspended  mass 
gives  the  correct  tension  of  the  wire.  Strike  the  fork  a 
single  sharp  blow  with  the  rubber  hammer,  place  its  stem 
on  the  sounding  box,  lightly  press  the  wire  down  on  the 
moveable  fret  and  pluck  the  wire  between  the  frets.  Alter 
the  position  of  the  moveable  fret  until  the  note  of  the  string 
is  nearly  in  unison  with  the  fork.  If  the  student  has  not 
a musical  ear,  he  can  find  this  position  of  approximate  unison 
by  placing  a small  rider  of  paper  on  the  wire  and  (without 
plucking  the  wire)  moving  the  free  fret  back  and  forth  until 
a position  is  found  where  the  rider  is  thrown  off  violently. 
The  wire  is  now  nearly  the  length  necessary  for  unison  of 
pitch  with  the  fork.  Strike  the  fork  again,  touch  it  to  the 
box  and  move  the  fret  backward  or  forward  a millimeter 
at  a time  noting  the  forced  vibrations  of  the  wire.  A po- 
sition will  be  found  where  the  “swell”  of  the  wire  is  a maxi- 
mum. Note  this  length  of  the  wire  as  the  length  for  unison 
with  the  fork.  The  swell  is  more  noticeable  when  the  vi- 
bration of  the  fork  is  vigorous,  hence  the  fork  should  be  struck 
and  tried  frequently.  A distance  of  a millimeter  to  either 
side  of  the  maximum  swell  length  will  cause  the  wire  to 
pulsate  (the  phenomenon  of  beats)  instead  of  vibrating  in  a 
single  prolonged  swell. 

By  this  same  process  determine  the  length  of  wire  in 
unison  with  the  second  fork.  Record  the  numbers  of  the 
forks  and  their  pitches,  also  record  the  lengths  of  wire  that 
vibrate  in  unison  with  them.  Test  with  the  data  found 
the  relation  that  pitch  varies  inversely  with  length. 

Relation  between  pitch  and  tension. — Determine  as  before 
the  length  of  the  smaller  wire  in  unison  with  the  first  fork 


288 


Manual  of  Experiments 


having  the  larger  mass  (about  4,000  gms.)  hung  to  the  free 
end  of  the  wire.  Determine  by  the  law  just  proved  the  pitch 
for  100  cm.  length  of  the  wire.  Repeat  using  same  fork  and 
both  iron  blocks  (about  6,000  gms.).  Calculate  the  pitch 
P for  100  cm.  length.  Having  thus  the  pitches  for  100  cm. 
lengths  of  wire  for  tension  of  4,000  and  6,000  gms.,  test  the 
relation  that  pitch  varies  directly  with  square  root  of  the 
tension. 

Relation  between  pitch  and  size  of  wire. — Using  small  wire 
and  tension  of  about  6,000  gms.,  determine  the  length  in 
unison  with  the  first  fork  and  calculate  the  pitch  P for  a 
length  of  100  cm.  Repeat  using  the  same  fork  and  tension, 
but  the  larger  wire.  Calculate  the  pitch  of  100  cm.  length. 
Note  on  the  small  card  attached  to  wires  the  mass  per  cm., 
then  test  the  relation  that  pitch  varies  inversely  as  the  square 
root  of  mass  per  cm.  of  length. 

Question. — In  what  way  do  your  results  depend  upon 
the  temperature  at  which  the  experiment  was  performed? 


55.  RESONANT  AIR  COLUMNS. 

(a).  To  determine  the  velocity  of  sound  in  air  by  reso- 
nant air  columns,  (b).  To  determine  the  pitch  of  a tuning 
fork.  (c).  To  determine  the  correction  factor  due  to  the 
disturbance  at  the  open  end  of  the  tube.  (D.  382-383; 
G.  278-279;  K.  308-309;  W.  316. 

Apparatus. — Resonance  apparatus,  two  tuning  forks, 
rubber  hammer. 

Description  and  Theory. — If  a tuning  fork  is  vibrated 
over  a column  of  air  its  vibrations  are  reinforced  and  made 
audible  to  the  ear  providing  the  column  of  air  is  of  the  proper 


In  General  Physics 


. 289 


length  to  produce  resonance.  Neglecting  the  correction 
due  to  the  disturbance  at  the  open  end  of  the  tube,  resonance 
first  occurs  when  the  length  of  the  confined  air  column  is 
equal  to  one-fourth  of  the  length  of  the  waves  set  up  by  the 
fork  used. 

, Resonance  will  again  occur  when  the  length  of  the  air 
column  is  further  increased  by  exactly  half  a wave  length. 

Let  X denote  the  wave  length  of  the  sound  produced  by 
the  fork,  h,  the  length  of  the  air  column  for  first  resonance 
and  k the  length  for  second  resonance,  each  measured  from 
the  mouth  of  the  tube;  then  for  first  resonance 

X = 4(/i+c)  (1) 

where  c is  the  correction  factor  due  to  the  disturbance  at 
the  open  end  of  the  tube.  For  second  resonance. 

X = 2(4-/1)  (2) 

The  velocity  of  sound  is  equal  to  the  number  of  vibra- 
tions per  second  times  the  wave  length,  or 

v~n\  (3) 

Combining  (2)  and  (3)  gives  for  the  value  of  v 

v = 2n(l2  — li)  (4) 

Equating  the  right  members  of  equations  (1)  and  (2) 
gives  for  the  value  of  the  correction  factor  c 

/2  — 3/i 

c = * ■ 5) 


According  to  Lord  Rayleigh  this  correction  should  be 
about  equal  to  the  radius  of  the  tube.  The  velocity  of  sound 
in  air  in  meters  per  second  is  given  approximately  by 

^t  = 331.7+.6 1 (6) 

where  t is  the  temperature  of  the  air. 


290 


Manual  of  Experiments 


The  resonance  apparatus  consists  of  a glass  tube  T, 
mounted  vertically  in  front  of  a scale  S,  Fig.  73,  with  a tuning 


fork  F clamped  above  the  open  end  of  the  tube.  The  water 
level  in  the  tube  can  be  varied  by  raising  or  lowering  the 
reservoir  R. 

Directions. — Clamp  one  of  the  tuning  forks  (the  one 
whose  frequency  is  given)  in  position  over  the  tube  as  shown 


In  General  Physics 


291 


in  Fig.  73.  Strike  the  fork  with  the  rubber  hammer  and 
raise  and  lower  the  water  level  in  the  tube  till  the  point  is 
found  where  the  tone  is  reinforced  most  strongly.  Measure 
the  distance  from  this  water  level  to  the  top  of  the  tube  and, 
if  it  is  the  first  point  of  resonance,  call  it  A Avoid  parallax 
when  taking  a reading.  Six  observations  should  be  taken 
for  the  position  of  each  node,  part  of  them  with  the  water 
rising  and  part  with  it  lowering. 

Lower  the  water  level  until  the  second  point  of  resonance 
is  obtained.  Determine  this  in  a manner  similar  to  the 
first  and  designate  it  as  /2.  Record  the  number  of  the  fork, 
its  frequency  and  the  temperature  of  the  room.  Substitute 
in  equation  (4)  and  determine  the  velocity  v.  Calculate  the 
theoretical  value  from  equation  (6).  Determine  the  percent 
error  in  your  results.  Also  determine  the  correction  factor 
c from  equation  (5). 

Clamp  the  second  fork  (frequency  unknown)  in  position 
and  determine  the  two  points  of  resonance  as  described  above. 
Substitute  in  equation  (4)  using  for  v the  value  obtained  from 
equation  (6)  and  solve  for  n. 

Questions. — (1)  Show  by  diagram  that  k is  one-quarter 
of  a wave  length  and  that  /2  is  three-quarters  wave  length. 

(2) — What  is  the  ratio  of  the  correction  factor  c to  the 
radius  of  the  tube? 


Appendix. 

I.  COMMON  CONVERSION 

Mass 

1 grain 
1 oz.  av. 
1 lb. 

Length 

1 centimeter  = 0.3937  inch  1 inch  = 

1 meter  = 3 . 280  feet.  1 foot  = 

1 kilometer  =0.6213  1 mile  = 

1 micron  = 0.001  mm.  1 mile  = 

p Area. 

1 sq.  cm.  = 0.1550  sq.  in.  1 sq.  in. 

1 sq.  m.  = 10.76  sq.  ft.  1 sq.  ft. 

Volume. 

1 cu.  cm.  = 0.0610  cu.  in.  1 cu.  in. 

1 cu.  m.  = 35.31  cu.  ft.  1 cu.  ft. 

1 liter  = 1.760  pints  1 quart 

• 

Angles. 

1 radian  = 57 . 295  degrees  1 degree 

Temperature. 

1°  C.  = (9/5) °F.  1°  F.  = 


1 gram  = 15.432  grains 
1 gram  = 0.0352  oz.  av. 
1 kg.  = 2.204  lbs. 


FACTORS. 


= 0.0648  gram.  / 
= 28.35  gram. 

= 0.4536  kg. 


2 . 539  cms. 
0.3047  meter. 
1 . 609  km. 
0.001  inch 


= 6.451  sq.  cm. 
= 0.0929  sq.  m. 


= 16.38  cu.  cm. 
= 0.02832  cu.  m. 
= 1 . 135  liters. 


= 0.01745  radian. 


(5/9)°  C. 


292 


II.  SPECIFIC  GRAVITY. 


Air,  at  0°  and  1 it- 
mo.,  .... 

.001293 

Alcohol,  at  20  . 

.789 

Aluminum 

2.6 

Brass  . • . 

8.5 

Copper  .... 

8.9 

Ether,  at  20° 

.715 

German  Silver  . 

8.5 

Glass,  common  . 

2.4— 2.6 

Glass,  flint  . 

3.  —5.9 

Hydrogen,  at  0° 
and  1 atmo 

.0000899 

Ice  . . . . 

.917 

Iron,  cast 

7.1— 7.7 

III. 

ELASTIC 

Iron,  wrought 

. . 7.8 

Kerosene  . 

.79 

Lead  .... 

. 11.37 

Mercury,  at  20°  . 

. 13.55 

Platinum  . 

. 21.50 

Quartz  . \ 

. 2.65 

Silver  . 

. 10.53 

Steam,  at  100° 

.00058 

Steel  .... 

. 7.8 

Sulphuric  acid 

. 1.832 

Turpentine 

.87 

Wood’s  metal 

. 9.7 

Wood,  oak 

.6— 

9 

Wood,  pine 

.4— 

7 

Wood,  walnut 

. .6— 

8 

CONSTANTS. 

Modulus  of  Elasticity 
or  Young’s  Modulus. 

Modulus  of  Rigidity  or 
Simple  Rigidity. 

dynes/sq.cm. 

lbs.  /sq.  in. 

dynes /sq.cm 

lbs.  /sq.  in. 

Multiply  by 

10 11 

10  6 

ion 

103 

Brass: 

- • 

cast 

6.5 

9 

2.4 

3.5 

wire 

10 

14 

3.7 

5.4 

Copper: 

, 

annealed 

10 

14 

— 

— 

cast 

12 

17 

4. 

6. 

wire 

12 

17 

4.5 

6.5 

Iron: 

annealed 

21 

30 

cast 

12 

17 

5.3 

7.6 

wire 

19 

26 

8.0 

12 

wrought 

20 

28 

7.7 

11 

Steel: 

Bessemer 

22 

31 

cast 

20 

28 

8.0 

12 

wire 

19 

26 

293 


IV.  MOMENTS  OF  INERTIA 


(m=mass  of  body,  l=length,  b=breadth.  r=external  radius,  n= 
internal  radius.) 


Body 

Axis 

Moment  of  Inertia 

Circular  disk) 

perpendicular  through 

or  cylinder/ 

center  of  flat  face 

J^mr2 

Ring  or  hollow) 
cylinder  / 

perpendicular  through 
center  of  flat  face 

+>  m (r2+ri2) 

Circ.  cylinder 

transverse  through  center 

m(r2/4+l2/12) 

Thin  rod 

transverse  through  end 

ml2/3 

Thin  rod 

transverse  through  middle 

ml2/12 

Rectangular/ 
block  / 

through  center  perpendicular 
to  face  with  sides  1 and  b 

m(l2+b2)/12 

Sphere 

through  center 

2mr2  /5 

Hollow  sphere 

through  center 

2 (r5 — r+) 

— m 

5 (r3 — r-,3) 

V.  BAROMETRIC  CORRECTIONS. 


Capillary  Correction— +0.031  cm. 
Temperature  Corrections 


Temp.  °F. 

Correction. 

Temp.  °F. 

Correction. 

60 

0. 171  cm. 

70 

0.225  cm. 

61 

0 . 176  cm. 

71 

0.231  cm. 

62 

0 . 181  cm. 

72 

0.236  cm. 

63 

0. 185  cm. 

73 

0.241  cm. 

64 

0 . 192  cm. 

74 

0 . 246  cm. 

65 

0.198  cm. 

75 

0.251  cm. 

66 

0.203  cm. 

76 

0.256  cm. 

67 

0.209  cm. 

77 

0.261  cm. 

68 

0.214  cm. 

78 

0.266  cm.  • 

69 

0.220  cm. 

79 

0.271  cm. 

Note — Temperature  readings  should  be  taken  from  the  Fahrenheit 
thermometer  on  the  side  of  the  barometer. 


294 


VI.  HEAT  CONSTANTS. 


Solids 


Substance 

Coefficient  of  Linear 
Expansion 

Specific  Heat 

Aluminum 

.0000255 

.219 

Brass  .... 

.0000189  - 

.090 

Copper  . . . 

.0000169 

.0936 

Glass  (crown) 

.0000078 

.161 

Glass  (flint)  . . : 

.000009 

.117 

German  Silver 

.000018 

.095 

Ic'e 

.0000507 

.502 

Iron  (cast) 

.0000102 

.11 

Iron  (wrought) 

.0000119 

.11 

Lead 

.0000276 

.031 

Nickel  ..... 

.0000128 

.109 

Platinum  . . . . 

.0000089 

.032 

Quartz  (fused) 

.00000039 

.174 

Silver 

.0000188 

.056 

Steel  (tempered)  . 

.0000132 

.12 

Tin 

.0000214 

.055 

Zinc 

.0000263 

.094 

Liquids,  Vapors  and  Gases 

(at  constant  pressure). 


Substance 

Coefficient  of  Cubical 
Expansion 

Specific  Heat 

Air 

.003671 

.237 

Alcohol  (ethyl) 

.00110 

.548 

Alcohol  Vapor 

.453 

Benzine  .... 

.00124 

.40 

Benzine  Vapor 

.299 

Ether  (ethyl)  . 

.00163 

.54 

Hydrogen  .... 

.003661 

3.409 

Kerosene  .... 

.0012 

.39 

Mercury  .... 

.0001818 

.0333 

Sulphuric  Acid 

.00059 

.34 

Turpentine 

.00094 

.420 

Water  (15° — 100°) 

.000372 

1. 

Water  Vanor 

.442 

295 


VII.  STEAM  TEMPERATURE. 

At  Different  Barometric  Pressures. 


h.cm. 

70 

71 

72 

73 

74 

75 

76 

77 

.0 

97 

72 

98 

11 

98 

50 

98 

88 

99 

26 

99 

63 

100. 

00 

100 

36 

.1 

97 

76 

98 

15 

98 

54 

98 

92 

99 

30 

99 

67 

100 

04 

100 

40 

.2 

97 

80 

98 

19 

98 

58 

98 

96 

99 

33 

99 

71 

100 

07 

100 

44 

.3 

97 

84 

98 

23 

98 

61 

98 

99 

99 

37 

99 

74 

100. 

11 

100 

47 

.4 

97 

88 

98 

27 

98 

65 

99 

03 

99 

41 

99 

78 

100 

15 

100 

51 

.5 

97 

92 

98 

31 

98 

69 

99 

07 

99 

44 

99 

82 

100 

18 

100 

55 

.6 

97 

96 

98 

34 

98 

73 

99 

11 

99 

48 

99 

85 

100. 

22 

100 

58 

.7 

98 

00 

98 

38 

98 

77 

99 

14 

99 

52 

99 

89 

100. 

26 

100 

62 

.8 

98 

03 

98 

42 

98 

80 

99 

18 

99 

56 

99 

93 

100. 

29 

100 

65 

.9 

98 

07 

98 

46 

98 

84 

99 

22 

99 

59 

99 

96 

100 

33 

100 

69 

VIII.  HYGROMETRY 


Pressure  of  aqueous  vapor,  p,  and  weight  of  water,  m,  contained  in 
one  cubic  meter  of  air  saturated  at  the  temperature  t°C. 


t 

P 

m 

t 

P 

m 

t 

P 

m 

—10 

cm. 

.22 

gm. 

2.4 

4 

cm. 

.61 

gm. 

6.4 

18 

cm. 

1.54 

gm. 

15.2 

— 9 

.23 

2.5 

5 

.65 

6.8 

19 

1.64 

16.2 

— 8 

.25 

2.7 

6 

.70 

7.2 

20 

1.74 

17.1 

— 7 

.27 

2.9 

7 

.75 

7.7 

21 

1.85 

18.2 

— 6 

.29 

3.1 

8 

.80 

8.2 

22 

1.97 

19.3 

— 5 

.32 

3.4 

9 

.86 

8.8 

23 

2.09 

20.4 

— 4 

.34 

3.7 

10 

.92 

9.4 

24 

2.22 

21.6 

— 3 

.37 

4.0 

11 

.98 

10.0 

25 

2.35 

22.8 

— 2 

.40 

4.2 

12 

1.05 

10.6 

26 

2.50 

24.1 

— 1 

.43 

4.5 

13 

1.12 

11.3 

27 

2.65 

25.5 

0 

.46 

4.9 

14 

1.19 

12.0 

28 

2.81 

27.0 

1 

.49 

5.2 

15 

1.27 

12.7 

. 29 

2.98 

28.5 

2 

.53 

5.6 

16 

1.36 

-13 . 5 

30 

3.16 

30.1 

3 

.57 

6.0 

17 

1.45 

14.4 

31 

3.34 

31.7 

296 


IX.  SPECIFIC  RESISTANCE. 


(Ohms  per  cm.3  at  0°C.) 


Aluminum 

3.  X10-6 

Manganin 

42.  X10-6 

Brass  

8. 

Mercury 

94. 

Copper  .... 

1.5 

Nickel  .... 

10. 

German  Silver  . 

20. 

Platinum 

8.9 

Gold  . . . . . 

2. 

Silver  . 

1.5 

Iron  ..... 

10.5 

Steel  .... 

50. 

Lead 

20. 

Zinc 

5.7 

X.  ELECTROCHEMICAL  EQUIVALENTS. 


Element 

Atomic 

Weight 

V alency 

Chemical 

equivalent 

Electrochemical 

equivalent 

Chlorine 

35.45 

1 

35.45 

.0003672 

Copper,  (cupric) 

63.6 

■ 2 

31.8 

.000329 

Hydrogen 

1.008 

1 

1.008 

.00001044 

Iron,  (ferric) 

55.9 

2 

27.95 

. 000289 

Iron,  (ferrous)  . 

55.9 

3 

18.49 

.000193 

Oxygen  . . 

16.0 

2 

8.0 

.00008283 

Silver  . . . 

107. 93 

1 

107.93 

.001118 

Zinc 

65.4 

2 

32.7 

.000338 

Nickel  .... 

58.6 

2 

29.3 

.000305 

XI.  INDEX  OF  REFRACTION  FOR  D LINE 

(>1=0.00005893) 


Air 

1 

000294 

Cedar  oil  .... 

1.516 

Alcohol,  ethyl  . 

1 

362 

Ether 

1.36 

Alcohol,  methyl 

1 

332 

Glass  (crown) 

1.51-1.62 

Benzine  .... 

1 

.504 

Glass  (flint)  . . . 

1 . 57-1 . 96 

Canada  Balsam 

1 

.54 

Quartz  .... 

1.54 

Carbon  bisulphide 

1 

.63 

Water  . . . ... 

1.334 

Notes  on  Tables  XII  and  XIII 

Table  XII — The  functions  of  angles  between  0 and  45°  (left  hand 
column)  are  marked  at  top  of  page,  those  from  45°  to  90°  (right  hand 
column)  at  bottom.  For  decimal  fractions  of  a degree,  add  pro- 
portional parts  of  the  difference  given  in  columns  next  to  functions. 

Table  XIII — Columns  on  right  of  page  give  numbers  to  be  added 
to  logarithms  for  the  corresponding  decimal  fractions  given  at  the  head 
of  these  columns. 


297 


XII.  NATURAL  TRIGONOMETRICAL  FUNCTIONS 


Degree 

Sine 

Tangent 

Cotangent 

Cosine 

Degree 

0 

0.0000 

0.0000 

0*0 

1 . 0000 

90 

1 

.0175 

175 

.0175 

175 

57.29 

0.9998 

02 

l 89 

2 

.0349 

174 

.0349 

174 

28.64 

.9994 

04 

88 

3 

.0523 

174 

.0524 

175 

19.08 

.9986 

08 

87 

4 

.0698 

175 

174 

.0699 

175 

176 

14.30 

.9976 

10 

14 

86 

5 

0 . 0872 

0 . 0875 

11.43 

0 . 9962 

85 

6 

.1045 

173 

.1051 

176 

9.514 

.9945 

17 

84 

7 

.1219 

174 

.1228 

177 

8.144 

.9925 

20 

83 

8 

.1392 

173 

. 1405 

177 

7.115 

.9903 

22 

82 

9 

. 1564 

172 

172 

.1584 

179 

179 

6.314 

801 

643 

.9877 

26 

29 

81 

10 

0.1736 

0.1763 

5.671 

0.9848 

80 

11 

.1908 

172 

.1944 

181 

5.145 

526 

.9816 

32 

79 

12 

.2079 

171 

.2126 

182 

4.705 

440 

.9781 

35 

78 

13 

.2250 

171 

.2309 

183 

4.331 

3/4 

.9744 

37 

77 

14 

.2419 

169 

.2493 

184 

4.011 

320 

.9703 

41 

76 

169 

186 

279 

44 

15 

0 . 2588 

0 . 2679 

3.732 

0.9659 

75 

16 

.2756 

168 

.2867 

188 

3.487 

245 

.9613 

46 

74 

17 

.2924 

168 

.3057 

190 

3.271 

216 

.9563 

50 

73 

18 

.3090 

166 

.3249 

192 

3.078 

193 

.9511 

52 

72 

19 

.3256 

166 

.3443 

194 

2.904 

174 

. 9455 

56 

71 

164 

197 

157 

58 

20 

0.3420 

0 . 3640 

2.747 

0.9397 

70 

21 

.3584 

164 

.3839 

199 

2.605 

142 

.9336 

61 

69 

22 

.3746 

162 

.4040 

201 

2.475 

130 

.9272 

64 

68 

23 

.3907 

161 

4245 

205 

2.356 

119 

.9205 

67 

67 

24 

.4067 

160 

.4452 

■""20  7 

2.246 

110 

.9135 

70 

66 

159 

211 

101 

72 

25 

0.4226 

0 . 4663 

2.145 

0.9063 

65 

26 

.4384 

158 

.4877 

214 

2.050 

95 

.8988 

75 

64 

27 

. 4540 

156 

. 5095 

218 

1.963 

87 

.8910 

78 

63 

28 

.4695 

155 

.5317 

222 

1.881 

82 

.8829 

81 

62 

29 

.4848 

153 

. 5543 

336 

1.804 

77 

.8746 

83 

61 

152 

231 

72 

86 

30 

0.5000 

0 . 5774 

1.732 

0.8660 

60 

31 

. 5150 

150 

.6009 

23o 

1.664 

68 

.8572 

88 

59 

32 

.5299 

149 

.6249 

240 

1.600 

64 

.8480 

92 

58 

33 

.5446 

147 

.6494 

245 

1.540 

60 

.8387 

93 

57 

34 

.5592 

146 

.6745 

251 

1.483 

57 

.8290 

97 

56 

144 

257 

55 

98 

35 

0.5736 

0.7002 

1.428 

0.8192 

55 

36 

.5878 

142 

.7265 

263 

1.376 

52 

.8090 

102 

54 

37 

.6018 

140 

.7536 

271 

1.327 

49 

.7986 

104 

53 

38 

.6157 

139 

.7813 

277 

1.280 

47 

.7880 

106 

52 

39 

.6293 

136 

.8098 

285 

1.235 

45 

.7771 

109 

51 

135 

293 

43 

111 

40 

0 . 6428 

0.8391 

1.192 

0 . 7660 

50 

41 

.6561 

133 

.8693 

302 

1.150 

42 

.7547 

113 

49 

42 

.6691 

130 

.9004 

3l  1 

1.111 

39 

.7431 

116 

48 

43 

.6820 

129 

.9325 

321 

1.072 

39 

.7314 

117 

47 

44 

.6947 

127 

.9657 

332 

1.036 

36 

.7193 

121 

46 

124 

343 

36 

122 

45 

0.7071 

1.0000 

1.000 

0.7071 

45 

Degree 

Cosine 

Cotangent 

Tangent 

Sine 

Degree 

298 


10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 


XIII.  LOGARITHMS. 


8 


4 5 6 7 8 9 


L523 

L818 


7007 

7093 

7177 

7259 


0128 

0170 

0212 

0253 

0294 

0334 

0374 

4 

8 

12 

17 

1 

21 

25  29  33  '37 
1 r 

0531 

0569 

0607 

0645 

0682 

0719 

0755 

4 

8 

11 

15 

19 

23  26  30 

34 

0899 

0934 

0969 

1004 

1038 

1072 

1106 

3 

7 

10 

14 

17 

21  24  28 

31 

1239 

1271 

1303 

1335  | 

1367 

1399 

1430 

3 

6 

10 

13 

16 

19  23  26 

1 

29 

1553 

1584 

1614 

1644  1 

1673 

1703 

1732 

3 

6 

9 

12 

15 

18  21  24 

27 

1847 

1875 

1903 

1931 4959 

1987 

2014 

3 

6 

8 

11 

14 

17  20  22  25 

2122 

2148 

2175 

2201  2227 

1 

2253 

2279 

3 

5 

8 

11 

13 

16  18  21  24 

* 

2380  2405 

2430 

2455  2480 

2504 

2529 

2 

5 

7 

10 

12 

15  17  20  22 

2625  2648  2672 

2695  2718 

2742 

2765 

2 

5 

7 

9 

12 

14  16  19  '21 

2856  2878  2900 

2923  2945 

1 

2967 

2989 

2 

4 

7 

9 

11 

13  16  18  20 

3075  3096  3118 

3139  3160  3181  3201 

t 

2 

4 

6 

8 

11 

13  15  17  19 

3284  3304  ^324 

3345  3365  '3385  '3404 

2 

4 

6 

8 

10 

12  14  16 

18 

3483  ,3502  3522 

3541  3560  3579  3598 

2 

4 

6 

8 

10 

12  14  15 

17 

3674 

3692  |3711 

3729  3747  3766  ,3784 

1 1 1 

2 

4 

6 

7 

9 

11  13  15  17 

1 

3856 

3874 

3892 

3909  3927  3945  3962 

2 

4 

5 

7 

9 

11  12  14  16 

4031  4048 

4065 

4082  [4099  4116  14133 

2 

3 

5 

7 

9 

10  12  14  15 

4200  4216 

1 

4232 

4249  4265  4281  4298 

2 

3 

5 

7 

8 

10  11  13  15 

'4362  4378  !4393 

4409  !4425 

4440  4456 

2 

3 

5 

6 

8 

9 11  13  44 

4518  4533  4548 

4564  4579 

4594  ,4609 

2 

3 

5 

6 

8 

9 11  12  14 

4669  4683  4698 

4713 

4728 

4742  4757 

1 

3 

4 

6 

7 

9 10  12  13 

1 

4814 

4829  4843 

1 

4857 

4871 

4886  4900 

1 

3 

4 

6 

7 

9 ,10  11 

13 

4955 

4969  '4983 

4997 

5011 

5024  5038 

1 

3 

4 

6 

7 

8 10  11 

12 

5092 

5105  5119 

5132 

5145 

5159  5172 

1 

3 

4 

5 

7 

8 

9 11 

12 

5224 

5237  5250 

5263 

5276  5289  5302 

1 1 

1 

3 

4 

5 

6 

8 

9 10 

1 

12 

'5353 

5366  5378 

5391 

5403  5416  5428 

1 

3 

4 

5 

6 

8 

9 10 

11 

5478  5490  5502 

5514 

5527  5539  5551 

1 

1 2 

4 

5 

6 

7 

9 10 

11 

5599 

5611  5623 

1 

5635  i5647  ,5658  5670 

1 

1 

2 

4 

5 

6 

7 

8 10 

1 

11 

5717  5729  '5740 

5752 

5763 

5775 

5786 

1 

2 

3 

5 

6 

7 

8 

9 

10 

!5832  5843  15855 

5866 

5877 

5888 

5899 

1 

2 

3 

5 

6 

7 

8 

9 

10 

5944  5955  5966 

5977 

5988 

5999  ^010 

1 

2 

3 

4 

5 

7 

8 

9 

10 

6053  6064  6075 

6085 

6096 

6107  6117 

1 

2 

3 

4 

5 

6 

8 

9 

10 

'6160  6170  6180 

6191 

6201  6212  '6222 

1 

2 

3 

4 

5 

6 

7 

8 

9 

,6263 

6274  6284 

6294  6304  6314  ,6325 

1 

2 

3 

4 

5 

6 

7 

8 

9 

6365 

6375  6385 

6395  6405 

6415  6425 

1 

2 

3 

4 

5 

6 

7 

8 

9 

6464  6474  6484 

6493  6503 

6513  '6522 

1 

2 

3 

4 

5 

6 

7 

8 

9 

6561  6571  ,6580 

6590  6599  6609  6618 

1 

2 

3 

4 

5 

6 

7 

8 

9 

16656  6665  6675 

l 1 i 

^684  6693  6702  ^712 

1 

2 

3 

4 

5 

6 

7 

7 

8 

6749  ‘6758  6767 

6776  '6785  6794  6803 

1 

2 

3 

4 

5 

5 

6 

7 

8 

6839  6848  6857  6866  6875  6884  6893 

1 

2 

3 

4 

4 

5 

6 

7 

8 

6928  6937 

6946  ,6955  6964 

1 1 

6972  6981 

1 

2 

3 

4 

4 

5 

6 

7 

8 

7016 

7024 

7033 

7042  7050 

7059  *7067 

1 

2 

3 

3 

4 

5 

6 

7 

8 

7101 

7110 

7118 

7126  7135 

7143  7152 

1 

2 

3 

3 

4 

5 

6 

7 

8 

7185 

7193 

7202 

7210  ,7218 

7226  7235 

1 

2 

2 

3 

4 

5 

6 

7 

7 

7267 

7275 

7284 

7292 

7300 

7308  7316 

1 

1 

2 

2 

3 

4 

5 

6 

6 

7 

17348 

7356  7364 

7372 

7380 

7388  7396 

1 

2 

2 

3 

4 

5 

6 

6 

7 

299 


Logarithms — Con. 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1 

2 

3 

4 

5 

6 

7 

8 

9 

55 

7404 

7412  7419 

7427  7435 

7443 

7451 

7459 

7466 

7474 

1 

2 

21 

3 

4 

T* 

O 

5 

6 

7 

56 

7482 

7490  7497 

7505 

7513 

7520 

7528 

7536 

7543 

7551 

1 

2 

2 

3 

4 

5 

5 

6 

7 

57 

7559 

7566  7574 

7582 

7589 

7597 

7604 

7612 

7619 

7627 

1 

2 

2 

3 

4 

5 

5 

6 

4 

58 

7634 

7642  7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

1 

1 

2 

3 

4 

4 

5 

6 

L! 

59 

7709 

7716  7723 

7731 

7738 

7745 

7752 

7760 

7767 

7774 

1 

1 

2 

3 

4 

4 

! 5 

6 

7 

60 

7782 

7789  7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

1 

1 

2 

' 3 

4 

4 

5 

6 

6 

61? 

7853 

7860  7868 

7875 

7882(  7889 

7896 

7903 

7910 

7917 

1 

1 

2 

3 

4 

4 

5 

6 

6 

62 

7924 

7931  7938 

7945 

7952  7959 

7966 

7973 

7980 

7987 

1 

1 

2 

3 

3 

4 

5 

6 

6 

63 

7993  8000  8007 

8014 

8021'  8028 

8035 

8041 

8048 

8055 

1 

1 

2 

3 

3 

4 

5 

5 

6 

64 

8062 

8069  8075 

8082 

8089  8096 

1 

8102 

8109 

8116 

8122 

1 

1 

2 

! 3 

3 

4 

_ 

5 

5 

^6 

65 

8129 

8136  8142  8149 

8156  8162 

8169 

8176 

8182 

8189 

1 

1 

2 

3 

3 

4 

5 

5 

6 

66 

8195 

8202  8209  8215 

82221  8228 

8235 

8241 

8248 

8254 

1 

1 

2 

3 

3 

4 

5 

5 

6 

67 

8261 

8267  8274  8280 

8287:  8293 

8299 

8306 

8312 

8219 

1 

1 

2 

3 

3 

4 

5 

5 

6 

68 

8325 

8331  8338  8344 

1 

8351 

8357 

8363 

8370 

8376 

8382 

1 

1 

2 

3 

3 

4 

4 

5 

6 

69 

8388  8395  8401  8407 

8414 

8420 

8426 

8432 

8439 

8445 

1 

1 

2 

3 

3 

4 

4 

5 

6 

70 

8451  8457  8463  8470 

8476  8482 

8488 

8494 

8500 

8506 

1 

1 

2 

2 

3 

4 

4 

5 

6 

71 

8513  8519  8525  8531 

8537  8543 

8549 

8555 

8561 

8567 

1 

1 

2 

2 

3 

4 

4 

5 

,5 

72 

8573  8579  8585  8591 

8597  8603 

8609 

8615 

8621 

8627 

1 

1 

2 

2 

3 

4 

4 

5 

5 

73 

8633  8639  8645  8651 

8657  8663 

8669 

8675 

8681 

8686 

1 

1 

2 

2 

3 

4 

4 

5 

*5 

74 

8692 

8698  8704  8710 

8716  8722 

8727 

8733 

8739 

8745 

1 

1 

2 

2 

3 

4 

4 

5 

5 

75 

8751 

8756  8762  8768 

8774 

8779 

8785 

8791 

8797 

8802 

1 

1 

2 

2 

3 

3 

4 

5 

5 

76 

8808 

8814  8820  8825 

8831 

8837 

8842 

8848 

8854 

8859 

1 

1 

2 

2 

*s 

3 

4 

5 

5 

77 

8865 

8871  8876  8882 

8887 

8893 

8899 

8904  8910 

8915 

1 

1 

2 

2 

3 

3 

4 

4 

5 

78 

8921 

8927  8932  8938 

8943 

8949 

8954 

8960  8965 

l 1 

8971 

1 

1 

2 

2 

3 

3 

4 

4 

5 

79 

8976 

8982  8987  8993 

8998 

9004 

9009  9015  9020 

9025 

1 

1 

2 

2 

3 

3 

4 

4 

5 

80 

9031  9036  9042  9047 

9053  9058 

9063 

9069  9074  9079 

1 

1 

2 

2 

3 

3 

4 

4 

5 

81 

9085  9090  9096  9101 

9106  9112 

9117 

9122 

9128  9133 

1 

1 

2 

2 

3 

3 

4 

4 

5 

82 

9138  9143  9149 

9154 

9159  9165 

9170 

9175 

9180  9186 

1 

1 

2 

2 

3 

3 

4 

4 

5 

83 

9191  9196  9201  9206 

9212  9217 

9222 

9227 

9232  9238 

1 

1 

2 j 

2 

3 

3 

4 

4 

5 

84 

9243  9248  9235 

9258 

9263  9269 

9274 

9279 

9284  9289 

1 

1 

2! 

2 

3 

3 

4 

4 

5 

85 

9294  9299  9304 

9309 

9315  9320 

9325 

9330 

9335  9340 

1- 

1 

2 

2 

3 

3 

4 

4 

5 

86 

9345  9350  9355 

9360  9365  9370  9375  938o' 

9385  9390 

1 | 

1 

2 

2 

3 

3 

4 

4 

5 

87 

9395  9400  9405  9410  9415  9420  9425  9430 

9435  9440 

0 

1 

1 

2 

2 

3 

3I 

4 

4 

88 

9445  9450 

9455  9460  9465  9469  9474  9479| 

9484  9489 

0 

1 

1 

2 

2 

3 

3I 

4 

4 

89 

9494  9499 

9504  9509  9513  9518  9523  9528! 

9533  9538 

0 I 

1 

1 

2 

2 

■ 

3 ; 

3 

4 

4 

90 

9542  9547 

9552  9557  9566  9566  9571;  9576 

9581  9586 

0 

1 

1 

2 

2 

3 | 

3 

4 

4 

91 

9590  9595 

| 

9600  9605  9609  9614  9619  9624 

1 ’ 1 r 1 

9628  9633 

1 

0 

1 

1 

2 

2| 

3 

3 

4 

4 

92 

9638  9643 

9647  9652 

9657  9661  9666  9671 

9675  9680 

0 1 

1 

1 

2 

2 

3 

3 

4 

4 

93 

9685 

9689  9694  9699 

9703  9708  9713  9717 

9722  9727 

0 

1 

1 

2 

2 

31 

3 

4 

4 

94 

9731 

9736 



9741 

1 

9745 

9750  9754  9759  9763 

III! 

9768  9773 

0 

1 

1 

2 

2I 

3 ! 

3 

4 

4 

95 

9777^  9782 

9786  9791 

9795 

9800  9805  9809  9814  9818 

0 

1 

1 

2 

2 

3! 

1 

3 

4 

4 

96 

9823  9827 

9832  9836 

9841 

9845  9850  9854  9859  9863 

0 

1 

1 

2 

2 

3 

3 

4 

4 

97 

9868 

9872 

9877  9881 

9886 

9890  9894 

9899  9903  9908 

0 

1 

1 

2 

2 

3 

3 

4 

4 

98 

9912 

9917 

9921  9926 

9930 

9934  9939 

1 

9943  9948  9952 

0 

1 

1 

2 

2 

3 

3 

4 

4 

99 

9956 

9961 

9965  9969 

9974 

9978  9983 

9987  9991  9996 

0 

1 

1 1 

2 

2 

3 1 

3 

3 

4 

300 


/ 


, 


